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scotianostra · 5 years ago

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On August 15th 1057 King of Alba, MacBeth, was killed at Lumphanan.

17 years and one day after he killed Duncan II to seize the crown Mac Bethad mac Findláich himself fell foul of the violent times. So what of his time as Monarch.......

We start today's anniversaries with the subject of my last post, I posted about this yesterday but we can still get more out of the story of the real MacBeth, ask most non-Scots to name a Scots king and they will eventually remember this fellow Macbeth, who murdered a kindly old man for his crown, egged on by his shrew of a wife who then went crazy and killed herself. Or you can tell them the snippets you pick up from the likes of this and others true stories about the real man.

While Macbeth lived, his name as a warrior-prince must have carried some weight among the other rulers of the countries within reach of Alba, his Scotland. Because of its situation between Scandinavia, England, Ireland and the continent, Alba was a place of strategic importance. In Macbeth, it seemed to find a capable and imaginative king who held the throne in disturbed times for 17 years and was able, indeed, to leave his shores for a very long time without fear of upheavals behind him – something his English contemporary, Edward the Confessor was never able to do, so it shows his, and perhaps, Scotland's standing on the continent of Europe carried more weight than our southern neighbours.

In fact, Macbeth went to Rome – an event about which Shakespeare knew nothing. We know the date – 1050 – from the chronicle of an Irish monk writing in Germany; and we know that Macbeth was free with his gold when he got there, scattering his alms ‘like seed’ To me this was some feat for a man from what was still perceived as a savage like nation like Scotland, or Alba, as it was still known as in those days. He may have visited Rome as a pilgrim. His reasons were more likely to do with the benefits a Roman association might offer a country backward in development, which had hitherto relied on the care and protection of the Celtic pastoral Church.

Macbeth and his kingdom stood at the hub of a power struggle in which the Norse and the Danes, the Holy Roman Emperor and the Saxons of England, the Normans and Flemings and the Celts of Brittany, Cornwall, Wales and Ireland all played a part, with the pope in Rome courting them all.

What of the eponymous real villain of the Shakespearean play, Lady MacBeth, well by all accounts she appears to have been a loyal and blameless lady. From a previous marriage, she brought him a stepson, Lulach, who Macbeth seems to have cherished, and who was crowned king after Macbeth, before being killed in his turn. Nor was she ever called Lady Macbeth. Macbeth, meaning ‘Son of Life’ or ‘of the Elect’ is not a surname. The king’s wife would have been addressed as the lady Gruoch in Gaelic. The name is recorded in Fife where she and her husband are said to have gifted land to the Celtic monks of St Serf’s island, Loch Leven.

And of course we can't talk about "The Scottish Play" as it is known in theatrical circles, as saying the name of the play is said to be back luck for those thespians, The Witches, well Hollinshead, he of the chronicles I spoke of yesterday, wrote about three women ‘in strange and wild apparel’, who promised Macbeth the thanedoms of Cawdor and Glamis as well as the throne, and who informed Banquo that his heirs, and not Macbeth’s, would rule Scotland. So you can see where Shakespeare stole the idea from. Don't hold that against the Englishman though, all the best storytellers get inspiration from other places, our own Walter Scott, Robert Burns and James Hogg the "Ettrick Shepherd" all used material gathered from ancient tales. Also the present castles of Glamis and Cawdor have no connection at all with this part of Macbeth’s story – indeed, there were no stone castles in mid-11th century Scotland, only halls and fortifications of wood. Nor can the ‘blasted heath’ and the ‘witches stone’ beside Forres be anything but inventions provoked by the legend.

If there were no prophecies, and no evil Lady Macbeth, why did Macbeth killed Duncan and Banquo, if not to seize the throne and prevent Banquo from founding the royal line? For this part we have to blame another man the Scottish philosopher and historian, and the first Principal of King's College in Aberdeen, a man called Hector Boece.

Boece comes in with the character of Banquo, Macbeth did not kill Banquo because Banquo did not exist. By the 15th and 16th centuries, when Boece was plying his trade, Scotland was a very different place to the one we see MacBeth in. The Stewars were in control, and on the surface they did not want to be seen as warriors, like our hero today, others like Duncan, to paper over the cracks in the story of Scotland's turbulent violent past, Boece used by James IV to paint the Scots in a better light, in doing so he greatly maligned the real Macbeth. The work was well received at the time, both in Europe and in Scotland. I'm not entirely sure about how Banquo came about, but that apart, by the time of the Stewarts, no one wanted to remember that once, kingdoms had to be ruled by men who were war-leaders, and thrones fell to the strongest and most worthy, and not automatically to the first-born. They certainly didn't want to remember that Alba may have been ruled by a bastard King, it disguised the fact that the line founded by Duncan sprang from an unorthodox marriage, to an abbot called Crinan. A murky business in Boece's time, best left alone!.

Finally we get to the end of the Shakespeare play and Birnham wood, where Macbeth was finally slain by the Scots lord Macduff, whose family Macbeth had caused to be murdered.

Again, in history, there was no MacDuff. The final fight in the play is based on The Battle of Dunsinane, Macbeth's army was defeated by Malcolm Canmore (Máel Coluim mac Donnchada) with the help of the aforementioned English King Edward the Confessor, It was said that MacBeth’s guard surrounded him and defended him as best they could but they were defeated 3000 – 1,500, but MacBeth eventually got away to safety and spent the next 3 years on the run. After Dunsinane Malcolm attended a meeting of the mormaers who elected MacBeth’s step son Lulach as king – perhaps thinking that MacBeth was dead – or that it was time to move on – who knows? Malcolm, of course, was not happy about this, seeing as he had fought really hard to win. He still had a support from the English soldiers, so he chased Lulach, if the king was not to be him, then he would kill whoever it was. And Lulach was caught and killed by Malcolm, another victim in the long line of murdered Scots kings. As far as killing their kings went, the Scots seemed to be pretty good at it, perhaps better than the English.

And so we reach the point of this post and three years after Dunsinane, MacBeth was finally cornered and killed at Lumphanan, his 17 year reign coming to an end. If there has to be a moral to this story it is ‘you will get yours eventually," it certainly mirrored Shakespeare in that way, if not others.

So Macbeth's Scotland may have very well been a place without towns or proper markets or roads, but his administration was clearly good for its time, however it needed later Norman-trained rulers and the help of the Church to develop what he had started and become a more peaceful nation, but remember, he wasn't the scheming power hungry monarch, egged on by an ambitious wife, but a good King who oversaw a a relatively peaceful time in our formation as a country.

#Scotland #scottisih #king #Monarchy #History #william shakespeare

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sonofhistory · 7 years ago

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didn't washington have not so good relationship with his stepson?

George Washington first met John Park “Jacky” in mid-March of 1758 when he courted his mother Martha Dandridge Custis. Jacky was on four years old at the time. While Washington adopted both Jacky and his sister Patsy, they retained their Custis surname. The children arrived to reside at Mount Vernon with their own slaves–Jacky’s was a ten year old named Julius. In his diary, Washington sometimes referred to his stepchildren as “Jacky Custis” and “Patsy Custis” as if they were visitors.

Although Washington enjoyed his children, “his formal presence tended to freeze their jollity.” His adopted grandson said, “They felt they were in the presence of one who was not to be trifled with.” He doted in Patsy, a girl who enjoyed music, while Jacky studied the violin and flute. He also hired a dancing master at Mount Vernon for the two children. Washington once himself stated that he had a more relaxed style with girls and that he could govern men not boys. Jacky shared many traits with his late father and the differences he had in temperament with his step-father created issues. Washington was harsh at the same time he was reluctant to apply discipline to Jacky.

Jacky was outwardly sweet and affectionate towards his mother but showed a degree of reserve and disrespect for Washington. His feckless nature was intolerable to Washington and he found himself in a predicament of wanting to disciple Jacky for his actions without seeming brutal to his wife. Towards the end of 1767, Washington sought to find a new teacher for the thirteen-year-old and contacted Reverend Jonathon Boucher, an Anglican clergyman who ran a small academy for wealthy boys in his home near Fredericksburg. In his introductory letter, Washington described Jacky as “as promising boy” who was “untwining in his morals and of innocent manners” but confessed his “anxiety to make him fit for more useful purposes than a horse racer.” Washington rode to Boucher’s school with Jacky, Jacky’s personal slave Julius and two horses.

Boucher’s first letter home described Jacky as a little angel, “a boy of so exceedingly mild and meek a temper” that Boucher worried he might be too artless, with “all the harmlessness of the dove” and none of “the wisdom of the serpent.” He concluded, “I have not seen a youth that I think promises ferried to be a good and useful man than John Custis.” However, a year later, Boucher whistled a different tune. “You will remember my having complained of Jack’s laziness, which, however, I now hope is not incurable,” he wrote to Washington. Later he wrote, “The chief failing of [Jacky’s] character are that he is constitutionally somewhat too warm, indolent and voluptuous.” He added, “Sunk in unmanly sloth, [Jacky’s] estate will be left to the management of some worthless overseer and himself soon be entangled in some matrimonial adventure.” Jacky saw little need to apply himself to his studies.

Martha Washington was a very overprotective mother who would not allow her son to swim because she feared he would drown. In the end, Jacky became so uncontrollable he began staying with other friends after his lessons and often spent the night elsewhere. “I would beg leave to request,” Washington told Boucher, “that [Jacky] may not be suffered to sleep from under your own roof … nor allow him to be rambling about at nights in company with those who do not care how debauched and vicious his conduct may be.” Boucher soon admitted he’d “Neve did in my life know a youth so exceedingly indolent to so surprisingly voluptuous. One would suppose nature had intended him for some Asiatic prince.” Boucher suggested the best way to control Jacky was to send his two horses back to Mount Vernon but Martha, his mother, refused.

Washington attempted to give his stepson everything education wise that he never got to have. He monitors Jacky’s education so narrowly because he took seriously his role as guardian of him. Early 1773, Washington found it high time to send Jacky off to college. For Martha, College of William and Mary would of been desirable considering it’s location, but Washington wanted to send Jacky off to College of New Jersey, however, Boucher wanted to steer him toward King’s College. After, Washington was decidedly in favor of King��s. In December of 1771, unfortunately, Jacky began to court Eleanor “Nelly” Calvert. Washington disapproved of the marriage but told Mr. Calvert to wait for marriage for two or three years until Jacky had completed his education.

When Jacky returned to Mount Vernon, he had with him in tow Charles Willson Peale who painted a portrait of Washington while there and miniatures of Martha, Patsy and Jacky. In May 1773, Washington accompanied his stepson to New York City and enrolled him in King’s College. Instead of socializing with other students, Jacky dinned with President Myles Cooper and his turbos. “I believe I may say without vanity that I am looked upon in a particular light” by the faculty, Jacky told his mother. “There is as much distinction made between me and the other students as can be expected.” He also bragged he and his friend Joe had their own suite rooms. When Patsy, his sister, died, his fiancee, Nelly, was staying with the Washington’s and her presence proved providential and she stepped into a huge emotional void left Patsy’s death, becoming like a second daughter.

During their first year of marriage, after Jacky dropped out of King’s College, Jacky and Nelly divided their time between Mount Airy and Mount Vernon, despite Martha’s wishes that they move permanently to Mount Vernon. Washington found solace that her son, of whom she was so attached to, may now provide emotional support for her and care for her. He asked Jacky and his bride to stay full time at Mount Vernon. While Washington was in Cambridge, on November 16th, 1775, accompanied by Jacky and Nelly, Martha piled into a carriage set for this destination. They arrived on December 11th, 1775 have not seen him since May. Washington soon pressed Jacky into service as a messenger.

In a letter in which Jacky intended to be solely about his mother’s recovery from her smallpox inoculation, he used the occasion to express gratitude for everything his step father had done for him, thanking him for the “parental care which on all occasions you have shown me. It pleased the Almighty to deprive me at a very early period of life of my father, but I cannot sufficiently adore His goodness in sending me so good a guardian as you, Sir. Few have experienced such care and attention from real parents as I have done. He is best deserves the name of father who acts the part of one.”

Martha, temporarily away from her husband at Mount Vernon, was there to witness the birth of her second grandchild to Jacky’s wife on New Year’s Eve in 1777. However, Jacky soon dwelled in selfish ways when he stalled in settling debts to his stepfather so he could repay in cheaper currency. With four children, Jacky took up residence at Mount Vernon in Washington’s absence and even named his last child George Washington Parke Custis in honor of him.

Jacky, in 1781 volunteered his services as an aide to Washington while before contributing only modestly to the war effort. Amid the unsanitary conditions at Yorktown, Jacky contracted camp fever and, knowing the condition most often to be fatal, expressed a last wish of witnessing Cornwallis’s surrender. He was carted thirty miles to Eltham in New Kent County, the estate of his uncle and his mother, his wife and his father-in-law were summoned to attend to him. By the time Washington arrived he learned the doctors had failed and Jacky was dying. The young man expired a few hours later, three weeks before his twenty-seventh birthday. On French observed remarked that Washington had a profound emotional response to Jacky’s death and was “uncommonly affected.” Washington remained in Eltham, attended to Jacky’s funeral before escorting his stepson’s recent widow and his wife.

Jacky left behind three small daughter and a baby boy. The Washingtons decided to adopt informally the two youngest children, Eleanor Parke Custis, then two years old called “Nelly” and George Washington Parke Custis, seven months old. Washington took seriously his duties toward the children and wrote in his will that it had “always been my intention, since my expectation of having issues has ceased, to consider the grandchildren of my life in the same light as I do my own relations.”

Jacky Custis left behind a murky legacy. Many years later his eldest daughter Elizabeth, raised by her mother and stepfather told how her father would hoist her on a table and forced her to sing indecent songs that he had taught her in order to divert his inebriated friends. “I was animated to exert myself to give him delight,” she wrote. “The servants in the passage would join in their mirth and I, hold my head erect, would strut about the table to receive the praises of the company. My mother remonstrated in vain.” Because he had not had a son until later, Jacky told his guests that little Elizabeth “must make fun for him until he had.”

#jack custis #John custis #flood pressles box #sincerely anonymous #Martha washington #George washington

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rap541 · 7 years ago

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A very Matthew and Mary Christmas to you from your Secret Santa at the Matthew Review!

(Gift receiver’s note- This was written for me by the lovely thematthewreview and promises to be continued! Also there’s DINOSAURS, which I adore! Such a lovely M/M Secret Santa gift!

DECEMBER 1998. Provost’s Hall, University of West Yorkshire.

Dr. Matthew Crawley, in his 11th year on the economics faculty at the University of West Yorkshire, had gotten a substantial bonus that Christmas from the school’s president, Dr. Robert Grantham. Dr. Grantham knew that somehow he was holding a brilliant scholar back from fulfilling his potential - he might have been setting public policy for Britain, but for a teaching post in a middling institution in the North, surely. But Matthew was a devoted family man as well as an academic, with a pretty wife, an eight year old boy, and two daughters, one four years old and the other a toddler just out of nappies.

‘You’ve had a busy year, Matt,’ Grantham said, clapping the younger fellow on the shoulder. 'Your book came out, you helped in the search process for Merriman over there. I hope this is enough for a nice, quiet vacation for you, Mary and the kids.’

'This is incredibly generous of you, though, sir,’ Matthew said, still reeling inside at the size of a cheque large enough to pay for a prudently managed two week trip to a resort somewhere.

But the elder knew it would be good for the younger, and spoke almost confidentially now:

'You’d like some time off, though, wouldn’t you, Crawley? Go ahead. Come back refreshed after spring break! Hug the kids, and say hallo to Mary, there’s a good chap. Now off you go home for Christmas. You have just one assignment, to go over holiday brochures.’

'I will, Robert. And thank you so much again!’

NEW YEAR’S DAY, 1998

'Rawr! RAWR!’ George made a little snarl, making a plastic tyrannosaurus rex chase a somewhat battered action figure (only slightly out of scale) around on the living room carpet. 'You’re going to be delicious! Rawwwwrrrr!’

Little Violet Crawley reached for some of the other brightly coloured new toys closest to her brother, knocking down a raptor striped in orange and red and knocking a puce green apatosaurus nearly a foot away, grabbing the grey brachiosaurus by its long neck. George turned his big eyes upon his sister, grabbed the raptor, and glared at her, bringing the trinkets in his hands to face her. He then raised his voice in a louder

'RRRRRAWWWWRRRRRRR!’

Violet began to cry, and little Belle woke up startled from a nap in her dozing mother’s arms. Dad had been watching, looking up every so often from the Financial Times.

'George Matthew Crawley, you apologize to your sister this instant,’ Matthew insisted in the low, stern tone the lad feared more than any animal noise.

'But, Dad, she took my dinosaur!’ came the boy’s petulant complaint.

'I mean it, George. I’d rather not take those away for a week so soon after Christmas Day,’ Matthew replied evenly.

'Oh, all right. I’m sorry, Violet.’

The little girl’s sobbing lessened.

'You pwomise?’

'I promise. You can play with my raptor, too, okay? It’s more fun with two.’

Matthew and Mary, pleased with that exchange, filed it in their memories to be discussed after bedtime came.

LATER THAT NIGHT

Now tuckered out enough to retire themselves about 2 hours after they had put the kids to bed, the handsome professor and his main squeeze got under the covers, and then he turned out the lights. Still definitely enough in love that sleepy kisses drizzled over several moments were their way of saying good night, they let themselves sink into each other’s arms, engulfed in Egyptian cotton and down.

'Matt?’ Mary spoke in a hushed tone.

'Yes, darling?’

'That was very sweet of George to let Violet play with his toys, wasn’t it?’

'It was, at that…’ he smiled, the warmth of his voice suggesting his expression in the darkness.

'Even if he’s been eating, sleeping and breathing dinosaurs ever since that movie came out…’

The one Matthew and Mary had to take their son to because a few of the characters got eaten by those animatronic beasts. They’d liked the approach to the story, though they would have liked more of the moral tale…

'Oh, I can understand raptors, though,’ came the husband’s sweet tone.

'You do, Dr. Crawley?’ his bride of nine years drawled in that appealing, familiar way that never failed to lead him on.

'Yes, indeed… I learned that the word “raptor” comes from the same Latin word as “rapture” does…’

'Yes, darling?’

'… to seize and sweep away. Like a peregrine, like a gyrfalcon, if you prefer.’

'How about like a snuggly creepy-Crawley? I like those kind best…’

'You catch on well…’ Matthew observed. 'Now then… rawwwwwwrrrrrrrr. Rawwwwrrrrrrrrrrr… oh, just raaaaawwwwrrrrrrrr, my darling.’

A honeyed sound, said in happy families like these to mean 'I love you’ in dinosaur. Ah, who is the raptor, and who their prey? It is the eternal question…

THE NEXT MORNING

Now, Matt having - eventually - slept on the wonderful possibilities of taking his family on holiday, he looked over some brochures as he drank his morning coffee at half past six, narrowing his choices down to the places that were the most relaxing, the most beautiful and with the most fun in it for the kids.

'Got the flyers out again, I see,’ Mary uttered as she brought dry toast, butter and marmalade to the kitchen table. 'Anything stand out from the others?’

'Oh, a couple of nice locations,’ Matthew replied. 'I’m thinking Dorset, Devon, maybe Cornwall; on a budget, of course.’

'That’s my Matthew. Oh, remember, by the way, school fees are due for George next week.’

'I have it on the calendar, darling.’

'When’s our holiday on the calendar, then?’

'In the middle of March, after the winter doldrums set in,’ he replied. 'Good time to get away…’

'And George will be on Easter break, and Violet still won’t have started school yet. Good,’ said Mary. 'We just need to make sure there will be enough for the kids to do.’

A FEW WEEKS LATER

Matthew had decided that they would go to the Heritage Coast, and stay in Lyme Regis, where the Easter Bonnet Parade, an aquarium, a museum, fossil hunting and the Cobb made up a list of major attractions. History and literature also played parts in his choice, as he thought it charming to see places associated with famous authors, Lord Admiral Nelson and the little collie, Lassie. The time was nigh to book a place to stay, and he found a small hotel in the northern part of town, the Seashell Inn. Children were welcome and ate breakfast for free. Why not? He rang up this Mr. Henry Talbot, the proprietor, and got an answer after a few rings.

'763904. Seashell Inn, Tony Gillingham here at your service….’

Easy enough to book a suite for a family of five, and Matthew was able to report to Mary as they did the dishes that night that the Crawleys now had a place to stay in Lyme Regis.

#matthew x mary #mary x matthew #m/m secret santa 2017 #submission

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riiroo · 5 years ago

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What To Teach Your Kids This September 2019

Well, it looks like we’ve seen the best of the summer days and now the kids are back at school. So, back to normal then :)

Each month at RiiRoo, we focus on different engaging topics to teach your kids in addition to what they would usually learn at school.

If you missed last month’s article - What To Teach Your Kids This August 2019, then you can read it here.

Last month we looked at Neil Armstrong’s birthday (the first man to walk on the moon), Play In The Sand Day (yes, this is a day), IBM PC announcement, Annie Oakley (sharpshooter) birthday, International Left Hander’s Day and National Aviation’s day.

This month

This month we’re going to look at Hitler invaded Poland; thus began World War II in Europe, The Great Fire of London, Treaty of Paris signed that ended the American Revolutionary War, Neptune Discovered and 9/11 - The worst terrorist attack in U.S. history.

1st September - Hitler invaded Poland

On September 1, 1939, under the rule of Adolf Hitler, Germany invaded Poland that triggered the worst war the planet has ever seen. World War ll. Believe it or not, Japan and China were also at war at the same time, so the world wasn’t a very nice place to live at this moment in time.

The Polish invasion lasted about a month before the Nazi victory. Unfortunately, this war plunged the world into a massive crisis and significant war that would last an astonishing six years and claim the lives of millions of innocent people on both sides.

Resources:

Here's Why That Move Marked the Beginning of WWII

Seventy-Five years ago, Hitler invaded Poland. Here's how it happened

Germany invades Poland

2nd September - The Great Fire of London

After so many people in London managed to avoid the Great Plague in 1665, they must have thought that the year 1666 would be a much better year. However, they were to be proved wrong.

In 1966 London witnessed one of the most massive fires the world has ever seen at the time.

The fire started in the King’s bakery in Pudding Lane near London Bridge.

Now, fires were quite common at the time and nothing out of the ordinary. The Lord Mayor at the time was woken up and told about the fire, in which he replied, “Pish! A woman might pi** it out!.”

Unfortunately, for London, they had a particularly hot summer that year, and there hadn’t been any rain for weeks.

Resources:

The Great Fire of London

In 1666, a devastating fire swept through London.

Ten things you (probably) didn’t know about the Great Fire of London

3rd September - Treaty of Paris signed that ended the American Revolutionary War

Even though this was a treaty signed in Paris, it was formed to end the American Revolutionary War in 1783.

The American statesman John Jay, John Adams, and Benjamin Franklin negotiated the peace treaty with Great Britain, and it was officially recognised by the Crown which brought about American Independence (as it is still known today).

It was down to the combined forces of the American and French troops that led George Washington to surround the British and its general Charles Cornwallis.

Resources:

The Treaty of Paris of 1783

Treaty of Paris (1783) - Wikipedia

Primary Documents in American History

23rd September 1846 - The planet Neptune was discovered

It’s strange to think that planets were discovered. I mean, they have been around forever, however, not for us humans.

We discovered the planet Neptune on September 23, 1846, using mathematics. Johann Gottfried Galle, Urbain Jean Joseph Le Verrier, and John Couch Adams worked independently, but all contributed to finding this new world.

Their work was in an international dispute as to whom to attribute the discovery since they all managed to discover it at the same time.

So What’s So Special About Neptune?

Neptune is the farthest planet we know of that is most distant from the Sun, so you could conclude that it’s pretty cold there. It is also the fourth-largest planet by diameter and 17 times the mass of earth.

Resources:

Today in science: Discovery of Neptune

Discovery of Neptune

Neptune’s discovery

11th September - The worst terrorist attack in U.S. history

A total of nineteen men committed the worst terrorist attack on US soil when they hijacked four fuel-laden US commercial planes destined for the Twin Towers, World Trade Center (WTC), The Pentagon and a Shanksville, Pennsylvania field.

In total, 2,977 people lost their lives in Washington DC, New York City and Shanksville, Pennsylvania.

The attack is said to have been orchestrated by al Qaeda leader Osama bin Laden in retaliation to US forces in occupying several oil-rich African countries.

Resources:

September 11 Terror Attacks Fast Facts

September 11 Attacks

FAQ about 9/11

29th September - World Heart Day

In May 2012, world leaders agreed that reducing global mortality from non-communicable diseases (NCDs) by 25% by 2025.

World Heart Day was created by the World Heart Federation to inform people around the world that Cardiovascular disease (CVD) can be reduced by having a healthy diet and reduce the consumption of alcohol and tobacco.

Resources:

World Heart Day

World Heart Day - Scale up prevention

So that’s it for another month. If you would like more interesting information like this, please sign up for the RiiRoo newsletter.

See you next month.

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wulfprints · 5 years ago

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SHOOT OF 2019-10-10

On the 10th of October, a friend and I headed to Cornwall to shoot some film. I took along my Nikon F100/50 mm F1.8 G for the same reasons outlined in my previous shoot report (portability, ergonomics, ability to create high-quality images in all light conditions).

This time I took a roll of Ilford HP5+. I wanted to experiment with black-and-white as I hadn’t in quite a long time. Black-and-white also has a much faster turnaround for me as I can hand-develop and scan it in uni in less than a day, unlike colour film which I have to take to a film lab (takes up to three days), and I wanted the images to be ready for the next group crit session on the day after (11th Oct), although this didn’t end up happening. I pushed the roll one-stop from 400 to 800 because from personal experience there’s really no reason not to. It gives you more light to work with and barely increases the grain. That being said, this might have come back to bite me.

The main reason it didn’t happen is that I felt very uninspired on the day. I think that I probably put myself under too much pressure to create images and this reduced my creativity. I find it quite difficult to produce 36 images in just one day without their quality dramatically decreasing as a result of the rush.

After viewing the negatives manually through a loupe I only ended up selecting four images to scan, and didn’t end up really liking any of them.

I also experienced very high levels of grain on this roll. This may have been because I pushed the film to 800 although I’ve done that many times before and never seen grain like this when doing so. It’s possible I agitated the film too much while developing it, I’ve seen that produce lots of grain in the past. It’s also possible the developing chemicals were a bit gone-off.

This is probably my ‘favourite’ image from the roll although I think that it would be better with a longer lens somewhere in the 100 mm range. I think that there is too much sky and sea here and it’d be nice to focus in closer on the central sailing boat. I do, however, like central compositions such as this. The ‘rule-of-thirds’ is, in my opinion, a bit pointless. I tend to just place the subject wherever I want to in the frame as I feel this is a more creative and natural approach. Quite often this results in a central composition, somewhat inspired by William Eggleston’s work.

I don’t like these two images at all to be quite honest. I think that, while I adore spontaneous and natural compositions normally, these just feel accidental and snapshotty. These look like holiday images to me, which is not something that I’m currently interested in producing. I don’t think that these images convey any real emotion to the viewer and they’re more documentary. I don’t think it helped that I was far out of my usual ‘shooting grounds’. Normally I like to shoot in the countryside yet here I was placed in the middle of a relatively highly-populated area. I also don’t think it helped that I drove around such a large area. This can make the shoot feel like a sweet-shop that never ends. The shot is never quite good enough because you feel you can always drive down the road to get a better one. I think it helps a lot to just park the car and walk so that you can feel the landscape rather than zoom through it.

This shoot was quite experimental and I didn’t really expect to get any final images from it to be fair. I do feel, however, that this was quite a painful roll of images that I didn’t really enjoy making, nor was it anywhere near my best work.

I don’t think I’ll try black-and-white again for a while as I don’t think it fits into the themes that I’m working from in my current project. I’m trying to convey the emotion of a scene and I think that removing the colour detracts from this. Humans don’t, after all, see in black-and-white, so it becomes a bit artificial and scientific.

Lessons learnt from this shoot: shoot more colour and less bw for the time being, focus more on immersing myself in one area rather than many areas, consider why grain was so intense on this roll, don’t ever rush myself to make images if I can help it.

#PHOT301

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wreadingandriting · 8 years ago

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My Potter Story, and Today

2 years ago, for Christmas, I was bought Harry Potter Trivial Pursuit. This was an excellent purchase, however I don’t think it was ever played as a family game. Or by the rules. My family more took turns picking questions they thought I wouldn’t get and groaning when I did. There weren’t very many. This was followed up by my mother finding a list of every character ever mentioned in the Harry Potter books and asking me who each and every one was. I missed only one or two, much to everyone’s eye rolling expectation. I don’t expect this article to be interesting for everyone, or indeed anyone. But I’d like my story to exist. I’d like people to see, if they want to, just what Harry Potter means to me

I remember getting my first Harry Potter book. It was in July 2000/2001. I was on holiday with my family in Cornwall, and had presumably run out of reading material. I convinced someone to buy me a book from the supermarket, and fatefully picked up Harry Potter and the Philosopher’s Stone. My God, I devoured that book inside of 24 hours (and at 6/7 years old that’s pretty impressive, if I do say so myself). I insisted on having the other 3 bought immediately, and hardly put them down over the following week. From that holiday, I was completely sold.

I don’t know how many times I’ve read them since. I have two sets, the originals (which are falling apart from the sheer number times they’ve been read by myself and other family members) and a new set of matching ones that I bought last year. Every time a new book came out, I got it at midnight and read it until I couldn’t keep my eyes open. My grandma would see me at lunchtime and ask how far I was and I’d inform her that I had already finished it. This was always greeted with a roll of the eyes.

As a child I remember buying the Comic Relief books, and putting them on a shelf with my Harry Potter books, covered with pieces of paper saying they were History of Magic or Herbology 101. I desperately wanted to get a Hogwarts letter. On my 11th birthday, nothing came, but I told myself it might not come until July or so. Like so many other children that year, I was disappointed. (I have found comfort in the knowledge that I could be a witch, but since the Muggleborn register was destroyed during Voldemort’s rise, I would never have received my letter.)

I played the Sorting Hat quiz and the wand game and all the other fun things that were on the Harry Potter Warner Bros website. I remember when the website went all dark and spooky the year that Prisoner came to cinemas. I played the Harry Potter and the Philosopher’s Stone game on my Playstation, and Chamber and Prisoner on the PS2. I still haven’t gotten past the Dementors on the Lake level. I completed the Pottermore website until it got it’s new update. I’ve been a Hufflepuff for years, and have recently convinced my entire family to find out theirs.

Harry Potter is one of only a few series that I have the ability to read time and time again, and never get bored. I watched all of the films in the cinema, and even had the first two on VHS. I loved the Cursed Child. I read it in a matter of hours, and even my partner (who says Harry Potter is “okay”) read it immediately after I put it down, and completed it in a similar timeframe.

Harry Potter to me means family. It means never being completely alone in the world. I love knowing that I have a connection with every person who’s ever read Harry Potter and found themselves hooked. I love knowing that Potterheads tend to be the most wonderful people, because we grew up with this sense of justice, and the triumph of good over evil. I love that all the Potterheads hate the very idea of Trump, because who votes for a man who is the living embodiment of Voldemort. I love the different theories people have about the Wizarding World. I love that JK Rowling apologies for a death every year.

I loved enthusing with my mum as a child over the books, and discussing the merits and flaws of the films. My sister is much younger than me, and as a result saw the films before she read the books. I remember gushing over which book she loved most, and how dare they not include S.P.E.W in the films, and wouldn’t Peeves have been so much fun.

Without Harry Potter, I wouldn’t love reading the way I do today. I would not have felt the excitement of waiting until midnight to buy something, stood with my dad in a street at 11:50pm. I wouldn’t have laughed and cried and worried and smirked and felt so triumphant. I wouldn’t have the strong sense of justice that I do today. I wouldn’t have moved on to loving Marvel and zombies and vampires. I wouldn’t have the friends I have today because I wouldn’t be the person I am today.

At the moment, I often think of a quote from the films. I am reminded of Dumbledore sat facing Harry when, for the boy in that moment, things couldn’t seem worse. “Dark and difficult times lie ahead. Soon we must all face the choice between what is right and what is easy.” Things may be different when I look back on this article. The world may be in a different place. I don’t know if that place will be good, bad, scary, or even here. But I also think of another quote: “We are only as strong as we are united, as weak as we are divided.” I think of all the Potter fans, and all the current protests going on across the world. I think of the solid unity and love that is felt throughout the planet against one person, one idea, one regime.

So to anyone reading this, who read any of the above and thought “I did that” or “I remember that”, don’t ever forget that Hogwarts is home to all of us, and that you are never alone. And never forget that there were good times, and there will be again. I promise.

~Ash~

#harry potter #potterhead #donald trump #today's world #wreading and riting #childhood #origin story #growing up potter #blog post #find me on facebook #find me on twitter #i like this blogging thing #books #fiction #fanfiction #writing #reading

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themartinsguide · 8 years ago

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Georgetown, Penang

9 - 12 January 2017

On our first morning in Penang we optimistically set our alarm for 7.30am in order to fit in a big day of sightseeing. However, despite being a very clean, chic, and happening place, The Frame Hotel is not particularly soundproof. With a room that overlooks the main street of central Georgetown, we may as well have been sleeping amongst the traffic (earplugs are one of the best innovations). So at 6.13am we were woken by the call to prayer resonating from the beautiful mosque positioned directly opposite our room. (The mosque is so close we have had to keep the shutters over our windows closed at all times due to James’ fondness of walking around nude. But that’s a different issue altogether, and an argument that Hannah has given up on. You win some you lose some). With time to kill we drank tea in bed and watched the sun rise and the temperature soar.

After receiving an annotated map and brief instructions from a very helpful receptionist, we headed for the Colonial Penang Museum (a Tripadvisor suggestion) which James cheerfully informed Hannah was 2km away. When we arrived at the doors of the Museum (a bit white colonial styled house) almost an hour later we were in agreement that the map was perhaps a “bit out of scale”. Although the walk through obviously one of the more affluent parts of time, where mansions lined streets was interesting, we arrived sweaty and irritable. So when we were told we had to take the guided tour to get access to the museum we acquiesced. Although as two employees on two separate occasions had told Hannah she was “beautiful” (and James was smart enough not to suggest that they said that to every female that turned up as a clever sales tactic) there was no way we weren’t going in. Our hopes were not high but in the spirit of not judging a book by its cover we paid the money and hoped for air conditioning.

Thankfully, our expectations were exceeded. While we didn’t consider it a “museum” in the strictest sense of the word, the house contained an impressive collection of antiques, artworks, and ornaments collected by the owner of the house over a number of years. While we could have done without the spiel about the porcelain doll collection, the Erard Piano Bukit by Francois Linke (circa 1905), of which there was only two ever built, was rather impressive. As were the two reverse paintings by William Morris & Co. But it was the huge tree hut that really did it for James. We agreed that the place would be a real hit with Jacqueline. She would love drinking G&T’s in that tree.

Following our hour (?!) long tour and our return hike (via the mall “just to check it out”) we were in desperate need of water, coffee and food. Victoria had recommended The China House for a good cup of coffee and she was right. The cakes and various baked goods also looked incredible (actually that seems to be a consistent theme all over Penang but we have, for the most part, been relatively restrained). After refuelling we went in search of Fort Cornwallis, just in case we hadn’t had enough history for one day. Unfortunately, for Hannah, this is when her need for a toilet struck quickly and viciously. We can only imagine how we looked as James, with chaffing in areas that caused him to walk in a limp (as our Fitbits told us we had covered 15km already), guided Hannah (cursing James and his ‘relaxed’ hygiene and iron stomach) through the busy, hot streets of Georgetown, making a beeline for our hostel. This wasn’t something Hannah was prepared to use a squat loo for. You get the picture.

Following a nap and a top up of air conditioner exposure, we decided to hit the local gym which Hannah discovered, much to her delight, did Les Mills Pump classes. Don’t roll your eyes. It’s actually quite fun trying out a new gym in a different country and this one had great views over the Penang hill.

In desperate need of a quick beer, and a hot curry to test Hannah’s stomach we headed out (after a shower). The curry was as one would expect - hot, quick, cheap and delicious; satisfied we wandered down Love Lane for a beer, or two. Love Lane is the centre of the backpacker ‘community’ (we jest) but is surprisingly pleasant and free from the Bintang singlet wearing mobs we expect to encounter in Thailand. While enjoying a quiet Carlsberg in Mickes Place, a rake-thin, deeply tanned chap with a Hawaiian shirt made a beeline for James (this is becoming a slightly worrying trend). Aussie Andy had arrived.

In the time it took him to order a beer Andy shared with us his concerns over the size of Malaysian adapters (and kindly offered us the one he was wearing as a necklace) and informed us that he had been waiting two days for his mate Raji to arrive in Penang. Two sips into a beer, and with the staff pleasantly ignoring his pesterings for a cigarette, Andy disappeared to buy a pack, but insisted that we hold his seat. Laughing with the waitress, who later confided in us that she was relieved when she realised we were Kiwis as we were likely to have a higher tolerance for Aussie Andy, we were quite confident that he was gone for the night.

We were wrong. 15 mins later Andy dashes past the entrance to the bar gesturing to two late model Mercedes where to park. Returning triumphantly Andy introduced us to Raji and Raji, one of whom was the owner of the local company that “ran security” for the majority of the bars in Penang and Butterworth. And had one eye. After a cursory round of chit chat One-Eyed Raji bought a round of drinks (which, as per “The Rules”, we accepted), but Andy was rather taken by a young woman and proceeded to decamp to her table. At this stage the chat really could/should have died off (as One-Eyed Raji didn’t speak English and Two-Eyed Raji preferred to stare and scowl), but James persevered, ignorant to Hannah’s slightly nervous fidgeting (luckily one of the staff had already surreptitiously checked to see that Hannah was OK). Preferring not to discuss how they knew Andy, and spying Raji Two Eyes’ interest with the football that was playing on the TV, the table launched into a spirited conversation about Premier League football. And so, a rather odd hour passed with our Malay acquaintances, interspersed with Andy pinching the band’s microphone and singing his original songs “Who’s got my twenty cents?” (Repeated in various melodies for 5-6 minutes) and “1-2-3-4-5-6-7-8-9-10-11-12” (it’s all in the name), occasional fist pumps and glass clinks with our new buddies, and conversations with two Austrian blokes who had been partying with Andy the night before and were a little surprised to see him alive. After Chelsea beat Peterborough (FA Cup 3rd Round) we graciously declined the Rajis’ offer to hit a club in Butterworth and headed home. Andy, go to bed.

A few too many beverages foiled our plan to wake early on the 10th and head out for a run. Not even the call to prayer could keep us awake and so after a lazy start we decided to hire a scooter (which James later insisted on calling his “Hog”) for the day. Hannah had not comprehended how terrifying this experience was going to be. Aware of James’ tendency to become easily distracted and having observed a serious and systemic lack of concern for road rules over the last few days, she was not impressed with the instructions to “just hold on”. She adopted the technique of just shutting her eyes and we made it to the National Park in one piece and decided to walk over to Turtle Beach (Hannah’s fondness for turtles meant that the alternative route to Monkey Beach didn’t stand a chance). The walk itself was quite steep and challenging, as the trail was irregularly maintained followed frequent washouts, but it was wonderful to get out of the city for a few hours. It was however slightly disappointing to get to Turtle Bay to discover no turtles (but definitely monkeys) and a no swimming sign, due to a savage undertow and venomous jellyfish. Returning back to the scooter (or James’ 'hog’) we set off with James under strict instructions to find food. This advice was taken loosely as he detoured for 30 minutes up a windy mountain road to “check out something”, which fortunately turned out to be a tropical fruit farm selling delicious smoothies and fresh fruit. A gentle cruise home followed, though Hannah would have preferred missing out on comments like “you might want to close your eyes, I think this is going to get a bit hectic”. Fortunately dinner and a cheeky gin was enjoyed without Andy re-appearing.

The 11th dawned bright and early with the call to prayer, and this time we did rise early for a 10km run along the waterfront before the heat got too intense. We had rather underestimated the fatigue from the previous two days walking though, and it turned into a rather quiet slog at a gentle pace. Still not learning our lesson we proceeded to hire bicycles and set off to bike/walk up Penang Hill. Hannah’s jean shorts were a very poor clothing choice though, and after pushing our bikes (which were not very well maintained, to say the least) straight up what seemed like the steepest street on earth for 20 minutes, while being stared at aggressively by monkeys, we called it a day and headed back to the hostel to relax, read, write and prepare to travel to Thailand tomorrow. Let the island-hopping commence!

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mathematicianadda · 5 years ago

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The Big Lock-Down Math-Off, Match 11

Welcome to the 11th match in this year’s Big Math-Off. Take a look at the two interesting bits of maths below, and vote for your favourite.

You can still submit pitches, and anyone can enter: instructions are in the announcement post.

Here are today’s two pitches.

Pat Ashforth & Steve Plummer – Maths or Art?

Pat Ashforth and Steve Plummer explore maths through knitting and crochet at woollythoughts.com.

In a previous match I talked about how perception might affect mathematical thinking. Here we are again with another example.

Are these the same?

Look at them with the eye of an artist and you will almost certainly say ‘No’. As a mathematician you might say ‘Yes’, or, at least ‘Maybe’.

You will definitely recognise this – or will you?

Most will instantly say ‘curves of pursuit’ – but is that right?

Technically, a pursuit curve is the path followed by the chaser in pursuit of the chased. This might be an animal pursuing its prey, one ship trying to catch another, or any other kind of chase. It is a single path. We usually see curves of pursuit where there are several chasers with each one pursuing the one in front and making pretty patterns. They don’t have to be in squares. Three equally-spaced chasers would make a triangle; six would make a hexagon. There can be any number of chasers. These are called cyclic or polygonal pursuits.

Right now we are only concerned with four chasers starting at the corners of a square field – but the squares you see above are not accurate cyclic curves. We have used artistic licence in our design. What you see are squares always rotated through the same angle. This is not what happens in a genuine curve of pursuit.

Assuming that all chasers are moving continuously at the same speed they will always cover the same distance in the same amount of time. The distance they move forwards will always be the same. This is not what happens in our design. The difference is not obvious on the first few squares but as you move further in you can see that the short side of the triangles remains the same throughout the ‘real curve’ but it is the angle that stays the same in the ‘artistic curve’.

The ‘real curve’ would be extremely difficult to replicate in knitting, as a separate set of instructions would be needed for each set of triangles. The knitted version uses the same shaping technique throughout, which results in triangles with equal angles, instead of a matching side. Our chasers are changing their speed, but are always chasing one another. You have probably noticed that the squares in our version don’t quite touch the edges of the previous square. We know that lines in a drawing, theoretically, have no thickness. Woolly things have quite thick joins.

The knitting starts with the centre square and each new square is four triangles added to the sides of the previous square. Having established that the rest should be straightforward. Not so.

The story of the knitting pattern goes back a long way. Briefly, it was first published in 1997 and hundreds of people made blankets. After a few years we started to get complaints from people who thought they had received the wrong pattern. They wanted the one with curves, not squares. We spent so much time explaining that what they saw depended on the colouring they used, we had to find a solution. Colours often cause problems because people feel the need to use exactly what was used in the original. The solution had to be in shades of grey. We made nine different versions so everyone could find what they wanted.

This is our original blanket which was made with rather uninspiring colours that we just happened to have at the time. We never expected this to still be our most popular design, more than twenty years later. Is it due to the maths or the art?

James Arthur – Functional equations

James is a confused physicist disguised as a pure mathematician who calls the heart of Cornwall his home. By day he is a web developer and studying towards his degree at Exeter Penryn. By night, he is a differential geometer and an analytical number theorist. He blogs at AlephJamesA.co.uk.

This story starts where all other good stories start: at an International Mathematics Competition. It shall be a story of intrigue, betrayal and in the end success.

IMO 2020 Jury Room

In the Summer of 2019 at Celtic Manor, a few miles from the Welsh Border. I was sitting in a conference room wearing my purple stellated icosahedron IMO t-shirt, sorting out bags to give to team leaders and other helpers at IMO. I was always a fan of the UKMT Challenges, and I had done the BMO2 that year and missed out on the Trinity Camp, but I was happy to have got so far. After all, I was at IMO anyway!

IMO stands for “International Mathematical Olympiad”.

I spent a lot of time in the room chatting and sorting stuff out, I also spent a lot of time running about Celtic Manor doing several bits and bobs. My main job was helping out in the Jury Meetings. I was a microphone runner, so I spent a lot of time getting to know the delegates and everybody else there.

Shhh… look over here

Now here we see a lesser spotted functional equation, it isn’t seen very often out of IMO in the UK. It is neglected by the education system and so is innately shy.

\[ f(2a) + 2f(b) = f(f(a+b)) \]

This problem is from Liam Baker, South Africa

It appeared on Day I of the 2019 Competition at Bath University. It was of much debate if I remember correctly, on whether it was too easy for the IMO.

How I came across it

While in a jury session, this problem took my interest. Functional equations had come into my find briefly in specific topics, especially when talking about Fibonacci Sequence and The Golden Ratio, but I had never studied them in great depth. I had just studied recurrence relations in excellent depth the term before.

I was sitting in the conference room working on a few problems that I was given by several delegates from other countries, they all took great interest in me, surprisingly! I was then handed a copy of this question, told to solve it and sworn to secrecy until later we found out it was on the paper. I had a problem, I have never solved anything like this before. Here is the exact problem:

Let $\mathbb{Z}$ be the set of integers. Determine all functions $f : \mathbb{Z}\to\mathbb{Z}$ such that, for all integers $a$ and $b$, $f(2a) + 2f(b) = f(f(a+b))$

… as in the paper, the version I was given was slightly different, but I believe that is still under embargo, so we shall go with this version.

While we are looking at this problem, we see that if $f(x) = 0$, then it would be a solution. We shall keep this in mind for later, and in particular, it’d be good to remember that if all the terms are in terms of $f$ then $f = 0$ is a solution.

The Road to Enlightenment

To solve this problem I was chucked through a crash course to solve these types of problems; I shall do a quick crash course for our purposes:

Types of Functions

Functional Friends

Three types of functions interest us: Injective, Surjective and Bijective. These really confused me, but I have got to grips with them now. They will be your little fluffy friends for Functional Equations. Friends that also are liable to bite the head off the problem, and so solve or it for you.

Def: Injective – Take a function from a set $A$ to a set $B$, $f: A\to B$, then if the values for $f(x)$ and $f(y)$ are equal, then $x = y$, i.e. $f(x) = f(y) \iff x = y$.

Def: Surjective – Take a function from a set $A$ to a set $B$, $f: A\to B$, then there is some elements of $A$ that satisfy $f(x) = b$.

Def: Bijective – Take a function from a set $A$ to a set $B$, $f: A\to B$, then it is sufficiently surjective and injective.

Learning By Example

The rest is a lot of plugging in and testing the outcome of substitutions. So let us take an example from IMO 2015, hosted at Chiang Mai, Thailand.

Let $\mathbb R$ be the set of real numbers. Determine all functions $f:\mathbb R\to\mathbb R$ that satisfy the equation $f(x+f(x+y))+f(xy)=x+f(x+y)+yf(x)$ for all real numbers $x$ and $y$.

Firstly let us look at this equation. We have a hanging $x$, we shall take this as a sign and let $x = 0$, letting $x$ and $y$ equal $0$ is usually a good idea to get an idea of what is going on.

\[ f(f(y)) + f(0) = f(y) + yf(0) \]

Now, we shall let $y = 0$ and simplify:

\[ f(f(0)) = 0 \]

Amazing, we have one piece of information, if we put $f(0)$ back into our function you get 0. So let us place $(0, f(0))$ back in using this information to try and get $f(0)$:

\[ f^3(0) + f(0) = f^2(0) + (f(0))^2 \]

Where $f^n(0) = f(f(f(…f(0))))$

Note that $f^{2n+1}(0) = f(0)$ and $f^{2n}(0) = 0$, so:

\[ \begin{align*}(f(0))^2 – 2f(0) &= 0\\f(0)(f(0) – 2) &= 0\\\implies& f(0) = 0 \text{ or } f(0) = 2\\\end{align*} \]

Now from this we shall let $f(0) = 2$ and come back to the other case later. Let us go back to the $x = 0$ substitution and let $y=x$:

\[ f^2(x) = f(x) + 2(x – 1) \]

We note the $x – 1$ and then we shall place a new substitution; $(x-1, 1)$:

\[ f(x – 1 + f(x)) = x -1 + f(x) \]

Now make the substitution of $x\to x-1+f(x)$

\[ f^2(x – 1 + f(x)) = f(x – 1 + f(x)) + 2(x – 2 + f(x)) \]

Which by (1) implies, $f(x) = 2 – x$

We have one solution, the second case is a bit more fiddly, it requires you to follow a similar argument and produces that $f(x) = x$.

Now, let us look at what we have done and denote some learning points:

Check if $f(x) = 0$ is a solution.

Aways start by substituting in (0, 0)

Look for non-functional expressions and try and make that zero and see if that leads anywhere

Take pieces of information gathered from above steps and plug into a new substitution, i.e. if you are looking for $f(0)$, then $(f(0), 0)$ and $(0, f(0))$.

and finally fiddle with substitutions until it produces a solution, or until you come to the fact you have an injection. Be persistent.

A quick note on injections

Injections are funny things, but they can be used to solve many different functional equations quickly. If you arrive at something of the form of: $f(g(\varpi)) = g(\varpi)$ (This is varpi, $\varpi$ is awesome!!). Then you can invoke an injection. This means that then if you have a second equation, make the substitution of $\varpi\to g(\varpi)$ and you have usually solved it.

To dust with the beast

The lesser spotted functional equation sits in its natural habitat soon to be unravelled by the curious mathematician, the mathematician stalks slowly towards the equation… he slowly brings out his substitutions for $a$ and $b = 0$, then pounces and sets $a = 0$, the equation squirms and transforms:

\[ f(0) + 2f(b) = f^2(b) \]

The equation reveals its secrets and quickly transforms back, that didn’t seem to work. Then the mathematician tries to net the creature with $b = 0$, the creature squirms again and transforms into a different form:

\[ f(2a) + 2f(0) = f^2(a) \]

This form seems to produce no new information and it transforms back. The mathematician curses and seems to be mesmerised by the ever so obvious non-symmetric $a$ and $b$, he wonders how to exploit that. He remembers that this has to be a function and spots out of the corner of his eye a darting linear motion from the beast, he tries to see if he can set the $f^2(x)$ to the same thing and cancel them, cutting the legs off the beast, making it easier to deal with. He lets $a = 0$ and $b = n+1$

\[ f(0) + 2f(n+1) = f^2(n+1) \]

Whilst taking hold of that transformed form, he then seizes an $1$ for $a$ and an $n$ for $b$ and pierces them into the beast.

\[ f(2) + 2f(n) = f^2(n+1) \]

He cuts the legs off the transformed forms and jams them together. Producing a difference:

\[ f(n+1) – f(n) = \frac{1}{2}(f(2) – f(0)) \]

He screams in triumphance as he realises that $f(x) = \alpha x + \beta$, plugging it back in he arrives at $f(x) = 2x + \beta$, where $\beta\in\mathbb{R}$, linearising the beast and turning it to dust.

To find out more about functional equations and other IMO maths stuff, I am to be writing more on my blog or if you want to read more about what life is like behind the scenes at IMO, read this.

So, which bit of maths made you say “Aha!” the loudest? Vote:

Note: There is a poll embedded within this post, please visit the site to participate in this post's poll.

The poll closes at 9am BST on Sunday the 3rd, when the next match starts.

If you’ve been inspired to share your own bit of maths, look at the announcement post for how to send it in. The Big Lockdown Math-Off will keep running until we run out of pitches or we’re allowed outside again, whichever comes first.

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primeetime · 5 years ago

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On September 11th

Kind of eerie to think how for every September 11th I’ll experience going forward, at some point, I am going to stop and reflect on what happened 18 years ago. It’s kind of eerie to think (knock on wood) that eventually I’ll probably have to tell my kids and grandkids about what happened on that day. It’s kind of eerie to think that one day would have so power and influence over my life that the events of that day will forever reverberate throughout my life until I die. But, ultimately that’s life. Previous generations had their “moments,” and our generation has ours. For me, September 11th will always be a day engulfed in tragedy. Normal everyday people woke up in the morning, went to work or the airport and never returned home. In an instant people and families who had clumsily tried to weave the fabric of their lives together found their lives ripped apart.

I remember being in school and seeing the number of my fellow classmates dwindle as frightened and scared parents stopped what they were doing and decided to focus on what mattered most. I remember my mom picking me up from school. I remember listening to the radio on the short ride from my school to my house and later watching the news and simply not being able to wrap my 9-year-old mind around the events of the day. I remember seeing the smoldering towers on TV and watching dark hazy objects fall from the windows. And once again not being able to wrap my 9-year-old mind around the fact that those dark hazy pixels on my tv screen were actually human beings made of flesh and blood like me. In short, September 11th was both the most terrifying and the most perplexing day of my childhood.

As I’ve grown older, however, I’ve come to realize that even though September 11th was a deep and dark tragedy that there were, even on that harrowing day, simple, beautiful, majestic acts of human courage that epitomize mankind at its best. I’ve come to read stories about firefighters and policemen who rushed into those burning towers, on the mere chance that they’d be able to save someone, ultimately putting their own lives at risk. I’ve read stories about the normal everyday people on united flight 93 who, upon hearing how other planes were hijacked, decided to fight to retake their plane. I’ve read stories like the one I share every year of Rick Rescorla…

“At 8:46 a.m. on the morning of September 11, 2001, American Airlines Flight 11 struck the North Tower of the World Trade Center (Tower 1). Rescorla heard the explosion and saw the tower burning from his office window on the 44th floor of the South Tower (Tower 2). When a Port Authority announcement came over the P.A. system urging people to stay at their desks, Rescorla ignored the announcement, grabbed his bullhorn, walkie-talkie, and cell phone, and began systematically ordering Morgan Stanley employees to evacuate, including the 1,000 employees in WTC 5. He directed people down a stairwell from the 44th floor, continuing to calm employees after the building lurched violently following the crash of United Airlines Flight 175 38 floors above into Tower 2 at 9:03 A.M. Morgan Stanley executive Bill McMahon stated that even a group of 250 people visiting the offices for a stockbroker training class knew what to do because they had been shown the nearest stairway.

Rescorla had boosted morale among his men in Vietnam by singing Cornish songs from his youth, and now he did the same in the stairwell, singing songs like one based on the Welsh song “Men of Harlech”:

“Men of Cornwall stop your dreaming, Can’t you see their spearpoints gleaming?, See their warriors’ pennants streaming, To this battlefield. Men of Cornwall stand ye steady, It cannot be ever said ye For the battle were not ready Stand and never yield!”

Between songs, Rescorla called his wife, telling her, “Stop crying. I have to get these people out safely. If something should happen to me, I want you to know I’ve never been happier. You made my life.” After successfully evacuating most of Morgan Stanley’s 2,687 employees, he went back into the building. When one of his colleagues told him he too had to evacuate the World Trade Center, Rescorla replied, “As soon as I make sure everyone else is out.” He was last seen on the 10th floor, heading upward, shortly before the South Tower collapsed at 9:59 A.M. His remains were never found. Rescorla was declared dead three weeks after the attacks.

Nothing will replace the lives that were tragically lost 18 years ago. But if all we remember, if all we focus on, if all we tell our children and our grand children and write in our history books are stories that leave out the simple, beautiful and awe-inspiring acts of courage that so many human beings engaged in and risked their lives doing on that fateful day we would be denying the deepest and most important truth of the human experience. That regardless of how bad things are and how dark things get, regardless of how tragic and stormy outer circumstances might be, that there are elements of the human spirit that embody and cling to the values of courage and love in the face of even the worst tragedy to occur in our lifetime, and those same elements exist in you and me.

“So courage is the power of the mind to overcome fear. Fear, unlike anxiety, has a definite object which can be faced, analyzed, attacked and endured. So often the object of our fear is fear itself. “Nothing,” says Seneca, “is terrible in things except fear itself.” And Epictetus says, “for it is not death or hardship that is a fearful thing, but the fear of death and hardship.” Courage can take the fear produced by a definite object into itself and thereby conquer the fear involved. “Courage,” says Paul Tillich, “is self-affirmation ‘in spite of’… that which tends to hinder the self from affirming itself.” It is self-affirmation in spite of death and non-being. He who acts courageously takes the fear of death into his self affirmation and acts upon it. This courageous self affirmation which is a sure remedy for fear is not to be confused with “selfishness.” Self-affirmation includes the right self-love and the right love of others. Erich Fromm has pointed out in convincing terms that the right self-love and the right love of others are interdependent, and that selfishness and the abuse of others are equally interdependent.

Courage is that quality which enables us to stand up to any fear. It is the final determination not to be stopped or overwhelmed by any object, however frightful it may be. Many of our fears are very real, and not mere snakes under the carpet. Trouble is a reality in this strange medley of life and dangers lurk beneath our every move. Accidents do occur and bad health stands as an ever threatening possibility. Death is a stark, grim and inevitable reality. We do ourselves and our neighbors a great disservice when we try to prove that there is nothing in this world to be frightened at. In this conundrum of life evil and pain are inescapable realities. The things that make for fear are close to all of us. These forces that threaten to negate life must be met and challenged by a daring “courage to be.” Courage is the power of life to affirm itself in spite of its ambiguities. It involves the exercise of a great and creative will. It is a bottomless resourcefulness that ultimately enables a man to hew out of the mountain of despair a stone of hope. Courage is the inner determination to go on in spite of obstacles and frightening situations; cowardice is the submissive surrender to the forces of circumstance. The man of courage never loses the zest for living even though his life situation is zestless; the cowardly man, overwhelmed by the uncertainties of life, loses the will to live. Courage breeds creative self‐affirmation; cowardice breeds destructive self-abnegation. Courage faces fear and thereby masters it; cowardice represses fear and is thereby mastered by it. So we must constantly build dykes of courage to ward off the flood of fear.”

“Love is the ultimate force that makes for the saving choice of life and good against the damning choice of death and evil. Therefore the first hope in our inventory must be the hope that love is going to have the last word.”

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Every UNESCO World Heritage Site in the UK

01 of 23

The English Lake District

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England's newest UNESCO World Heritage Site covers more than 885 square miles of Cumbria in the northwest corner of England, just below the Scottish Border. The region includes more than 50 lakes and mountain tarns as well as England's highest mountain, Scaffell Pike, and three others of more than 3,000 feet.

When the railroads arrived in the area in 1840, the Victorians followed and this became the first part of Britain that saw organized touring and vacation travel.

Unsuited for most kinds of farming, the Lake District became one of England's primary areas for raising sheep. The needs of sheep and sheep farmers, in turn, shaped the landscape. Among those who sought to retain the Lake District's harmonious way of life was children's author Beatrix Potter, who lived, farmed and wrote here. During her lifetime, she bought up thousands of acres of farms and pastures. When she died, she left them, along with a considerable fortune, to the National Trust.

The Lakes and the Lakeland Fells have, over the years, inspired many more authors, from one of the earliest female travel writers and diarists, the intrepid Celia Fiennes in 1698, through a host of Romantic poets—Wordsworth, Samuel Taylor Coleridge and Robert Southey along with their visitors, Shelley, Sir Walter Scott, Nathaniel Hawthorne, Keats, Tennyson and Matthew Arnold.

02 of 23

The City of Bath

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From its 2,000 year old Roman Baths to its Georgian terraces and Pump Room, the entire city of Bath was listed by UNESCO in 1987, one of the earliest world cities to be inscribed on the World Heritage List.

The Roman baths and the temple complex together with the remains of the Roman city, Aquae Sulis, are the most famous and important Roman ruins north of the Alps. They are one of only a small number of Roman bath complexes worldwide actually heated by natural hot springs (the only hot springs in Britain).

The Palladian architecture of the 18th century spa town, developed during the reign of George III, incorporates and preserves the Roman site in their layout and design.

Jane Austen enjoyed the health giving waters of Bath though she didn't think as much of the accompanying social scene and marriage market as did many of her characters. Besides its feast of historic architecture, Bath has great restaurants, top shopping, quirky museums, a lively cultural scene and brand new in the 21st century, a multi-million pound, thermal spa and a new luxury hotel with the hot spring waters actually pumped into guest rooms.

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Blaenavon Industrial Landscape

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In the 18th and early 19th century, Blaenavon in South Wales was one of the world's largest suppliers of coal and iron manufacture. The iron foundries and the coal mines that originally put Blaenavon on the map still remain.

Blaenavon was inscribed on the list in 2000 in recognition of its demonstration of the dynamic forces that shaped the early industrial revolution. Today, visitors can descend deep into the ground at The Big Pit, Wales National Coal Museum,. This was the last deep working coal mine in the area and when it closed in 1980 it ended and era that began with the Blaenavon Iron Works around 1789. The ironworks are considered the best preserved 18th century example in the world. The existing site includes remains of the late 18th and 19th century furnaces, original casting houses and kilns, workers’ housing, a massive chimney,cast-iron pillars and brackets and a water balace tower that demonstrates early life technology using water to counter-balance loads.

The nearly 13-square-mile site is laced with self-guided walks across a valley loaded with evidence of early settlement and industry.

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Blenheim Palace

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The only palace not in Royal hands in England, Blenheim Palace was a gift from Queen Anne to John Churchill, the first Duke of Marlborough and ancestor of Winston Churchill—who was born there. The grant was in recognition of his military victory at the Battle of Blenheim. The 18th century house, built between 1705 and 1722 by John Vanbrugh and Nicholas Hawksmoor, is set in a 2,100 acre park, designed by Capability Brown. Among Brown's achievements are lakes and a fabulous cascade that looks like a natural waterfall but it totally down to Brown's skill and artiface. Wander around the park and you might just spot the current Duke, who still occupies part of the house.

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05 of 23

Canterbury Cathedral, St Augustine’s Abbey, and St Martin’s Church

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Considered the “Mother Church” of the Anglican Communion, Canterbury Cathedral dates it's origins to St. Augustine, sent to convert the Britons more than 1400 years ago. The ruins of St. Augustine's Abbey, just outside the city walls, (which you can explore with VR goggles) date from AD 597. The Cathedral is also where St. Thomas à Becket was martyred after a possibly offhand remark by King Henry II. The King and Becket (who was then Archbishop of Canterbury and had been the King's boyhood friend) argued over whether the King's law took precedence over church law. Henry was heard to remark, “Will no one rid me of this troublesome priest” and soon armed knights attacked Becket with swords while he knelt in prayer in the Cathedral. The spot is marked by a candle to this day. Chaucer's pilgrims were headed here in The Canterbury Tales.

St Martin's Church, a parish church founded sometime before 597AD, also included in this World Heritage Site, is the oldest church in continuous use in the English speaking world.

In addition to the Cathedral and Cathedral Precincts, Canterbury is well located in Kent for visitors to coastal locations such as Whitstable, Chatham and Rochester.

06 of 23

The Castle and Town Walls of King Edward in Gwynedd

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If you're a history buff, you'll need to travel around North Wales to see King Edward I's ambitious building program designed to cow the Welsh into recognizing him as their king.

Edward I of England led two military campaigns against the Welsh in the late 13th century. Eventually, he surrounded the North Wales province of Gwynedd with castles. These castles and fortified complexes—Beaumaris, Harlech, Caernarvon and Conwy—designed by his architect James of St. George, are considered the finest examples of 13th and 14th century military architecture in Europe.

07 of 23

Cornwall and West Devon Mining Landscape

If you've been following the BBC series Poldark, you'll recognize the characteristic engine house of Poldark's ever struggling tin and copper mine, Wheal Leisure. What you may not know is that in the 18th and 19th century, Cornwall and West Devon dominated the world supply of copper and tin. Copper was in demand to clad the hulls of the British Empire's wooden ships; from Napoleonic times onward, tin demand grew for the canning of food. The technology used in this part of southwest Britain led the world.

Today, this World Heritage Site, inscribed in 2006, is divided among ten different locations within close proximity of each other, protecting engine houses, beam engines, technology, transportation and communities important to this industry between 1700 and 1914.

Several of the mines used as locations in Poldark are within the World Heritage Site and can be visited. They include:

Botallack in St Just

Paths at Wheal Charlotte, Wheal Coates or Trevellas, in the National Trust site at Chapel Porth.

Levant Mine and Beam Engine, St Just.

You can also take an underground guided tour at Poldark Mine, the only complete tin mine in Cornwall open to visitors.

08 of 23

Derwent Valley Mills

Glen Woodccl

For good or ill, the factory system was born here when entrepreneur Richard Arkwright adapted and enlarged an earlier invention, the spinning jenny, into the water powered “spinning frame” and created an industry. His invention enabled mass production of strong cotton yarn and Britain's cotton textile production on a world conquering scale was born. Arkwright's 18th century model factories created a template that spread around the world. The mill buildings of New England, especially those beside the river in Lowell, Massachusetts, were influenced and inspired by Arkwright's Derwent Valley factories.

Because later development of the factory system moved on to urban settings, several the mills and mill communities here remained relatively unchanged for centuries.

The valley of the River Derwent lies near the eastern edge of the Peak District National Park in Derbyshire. Among several historic buildings that can be visited at this World Heritage site, Richard Arkwright's original 1783 cotton mill, Masson Mills, is a highlight. The earlier Cromford Mills nearby, built by Arkwright in 1771, was the world's first successfully water powered cotton mill.

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09 of 23

Dorset and East Devon Coast

You've heard of Jurassic Park no doubt, but did you know that England has a real Jurassic Coast? It's 95 miles of the East Devon and Dorset Coast, in Southwest England. About a third of it is owned and protected by the National Trust. It's composed of wild beaches, sheer white cliffs and stunning rock formations. Important (and easily seen) evidence of the history of life on earth—185 million years of it—is included at this site.

10 of 23

Durham Castle and Cathedral

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A BBC poll choose Durham Cathedral as Britain's best-loved building. Built in the 11th and 12th centuries to house the relics of St. Cuthbert, evangelizer of Northumbria, and historian The Venerable Bede, it has been in continual use and occupation for 1,000 years.

The castle, behind it on a peninsula, is an ancient Norman fortress that housed the prince-bishops of Durham. Today it is part of Durham University and, amazingly, you can book a room to stay there. But visits to the Castle are by guided tour only, so check their website to book.

11 of 23

Frontiers of the Roman Empire

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This is a multi-national site that reflects the northernmost extent of the Roman Empire in the 2nd century AD. Part of this UNESCO World Heritage inscription stretches across nothern Germany.

In the UK, there are two important areas:

Hadrian's Wall : As the Roman Empire began to crumble, the Romans built a defensive wall, across the North of Britain, from Carlisle to Newcastle-on-Tyne, with further fortresses on the west stretching south along the Solway Firth. Today, remnants of the wall can be found for about 73 miles. Excavations at Vindolanda, a fort and village on Hadrian's Wall, provide a glimpse into the life of a Roman legion at the edge of the empire. Exhibitions include rare letters home and are among the only examples of Roman handwriting in the world. Hadrian's Wall has been inscribed in the world list since 1987.

The Antonine Wall: Twenty years after Hadrian built his wall, in 142 AD, the Emperor Antonius Pius attempted to extend the empire 60 miles further north and built what is now known as the Antonine Wall. Traces of it – some the stone foundations of mile-castles and some little more than ditches or embankments, reach across Scotland from the Firth of Clyde to the Firth of Forth. This evidence of the Roman Frontier was added in 2008.

12 of 23

Giant’s Causeway and Causeway Coast

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The Giant's Causeway, near Bushmills on the North coast of County Antrim, Northern Ireland, isn't man made. Northern Ireland's only UNESCO World Heritage Site may look like a roadway into the North Atlantic but it is one of Ireland's natural phenomena, made of about 40,000 interlocking, hexagonal basalt columns. They are the remnants of an ancient volcanic lava glow, frozen in time—some more than 12 meters high. The tops of the columns form stepping stones, mostly hexagonal (six-sided) but also with four, five, seven and eight sides, leading from the foot of a cliff into the sea.

The Causeway is just part of the Causeway Coast that also includes the terrifying Carrick-a-Rede rope bridge; Northern Ireland's tallest cliff face; Dunseverick Castle, where a waterfall drops straight into the sea; and the haunted ruins of Bonamargy Friary.

The visitor's center, opened by the National Trust, brings the science, the history and the great Irish legends and stories associated with the causeway and coast to life.

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13 of 23

The Heart of Neolithic Orkney

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Visitors to Orkney are immediately struck by the enormous concentration of mysterious prehistoric structures that dot the islands. Some are more than 5,000 years old, predating Stonehenge and the Pyramids by several thousand years. The site includes two very different stone circles, The Standing Stones of Stenness and The Ring of Brodgar. There's also a chambered burial mound called Maeshowe, full of Viking runes from a later period, and a 5,000 year old village, Skara Brae, with a number of un-excavated mounds and sites.

14 of 23

Ironbridge Gorge

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A huge number of early industries gathered around this strikingly beautiful river gorge in rural Shropshire in the late 18th century. Soon, contemporaries described it as “the most extraordinary district in the world” and “the birthplace of industry.” With its 18th century furnaces, factories, workshops and canals, and the world's first iron bridge, the site continues to excite visitors.

15 of 23

Liverpool: Maritime Mercantile City

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Famous, of course, for The Beatles, on a more sober note, Liverpool's early fortunes were made in international trade. Its role in the the slave trade makes it a moving and important place to visit for anyone interested in this aspect of history.

Liverpool is currently on the “List in Danger” because of controversial developments planned nearby.

16 of 23

Maritime Greenwich

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If you've heard the phrase “Greenwich Mean Time” then you know one of the reasons this ensemble of buildings enclosed in a 17th century park is important. The Royal Observatory engaged in the early astronomical work that made modern navigation possible. Observations by Robert Hooke, and John Flamsteed, the first Astronomer Royal, assured the first accurate measurement of the earth’s movement contributing to accurate global navigation. Today, when you visit the observatory you can straddle 0º longitude and learn about the base-line for the world’s time zone system.

Other buildings on the site include the first Palladian building in Britain, the Queen's House, designed by Inigo Jones; the Royal Hospital (now part of the University of Greenwich), a collection of Baroque buildings designed by Christopher Wren and Nicholas Hawksmoor, and parts of Greenwich town center. The Royal Park, popular with visitors and locals and the site of equestrian events in the 2012 Olympics, was designed by André Le Nôtre in 1660.

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17 of 23

The Palace of Westminster, Westminster Abbey and St Margaret’s Church

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Westminster Abbey is where English monarchs have been hatched, matched and dispatched for nearly 1,000 years. In other words, it has been the location of coronations, royal weddings and royal funerals (though not often burials) for centuries. King Edward the Confessor spent so much time establishing the Abbey that he neglected to have an heir, opening the door to the Norman Conquest. He's buried in the Abbey and his successor, William the Conqueror was crowned here.

Next to the Abbey, The Palace of Westminster—called the Mother of Parliaments—is a 19th century gothic revival on the footprint of Edward's original palace—some of which remains deep within the building. And nestled between the two and dwarfed by them, St Margaret's Church was created in the Middle Ages to serve the people of Westminster so they wouldn't disturb the Benedictine monks, who then controlled the Abbey, at their prayers.

Together, these three buildings represent nearly about eight centuries of architectural development and the relationship of the monarchy, civil power and the church in forming modern Britain.

18 of 23

Pontcysyllte Aqueduct and Canal

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The virtually unpronounceable (unless you speak Welsh) Pontcysyllte Aqueduct carries the Llangollen Canal across the River Dee at a height of 126 feet. At only 11 feet wide—just about the width of an English narrow boat with inches to spare on either side—this can be a hair-raising 1,007 foot long journey for anyone worried about heights.

The canal, used by thousands of narrow boat enthusiasts every year is 204 years old and recognized by UNESCO in June 2009 as the masterpiece of pioneering 17th and 18th century civil engineer Thomas Telford, one of the modern world's earliest and greatest bridge, road and canal builders.

In 2012, the Olympic Torch was carried across the canal in a narrow boat on its journey around Britain. Volunteers in Victorian dress towed the boat across. But don't worry. If you decide to do a narrow boat tour on the Llangollen Canal, you can hire a motored boat that crosses at a walking pace. Or join a public cruise across, try a horse-drawn narrow boat or even canoe across. But don't look down.

19 of 23

Royal Botanic Gardens, Kew

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This 300 acre garden on the western edge of London in Kew ( a village of the Royal Borough of Richmond), claims the “largest and most diverse botanical and mycological collections in the world”. Started as a Royal garden in 1759 on the site of an earlier exotic garden, it became a national institution in 1840.

The site includes 44 listed buildings including several historic, iron framed glasshouses. The gardens hold more than 30,000 living plants as well as at least seven million preserved specimens. In addition to being a worldwide research center for the study of plants, conservation and ecology, Kew also demonstrates garden artistry and design over more than 250 years. Easy to reach by London Underground or bus from Central London, Kew is wonderful to visit at any time of year.

20 of 23

Saltaire

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Textile mill owner and philanthropist Sir Titus Salt created Saltaire as a complete community for his workers in the 1850s. The village is named for Salt and for the River Aire, in West Yorkshire near Bradford, on which it is located.

Mills, the employees' housing, the dining room, Congregational Church, almshouses, hospital, school, institute, and a park all still remain and many are still in use. The World Heritage site demonstrates the emerging paternalistic concern of Victorian employers for social welfare, health and education of their workers. It served as a model for the “garden city” movement in Britain, the USA and elsewhere.

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21 of 23

Stonehenge, Avebury and Associated Sites

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No one knows who built Stonehenge, as much as 5,000 years ago, or why they did it, but Britain's most iconic sight has captured the imagination of of visitors for tens of centuries. Nearby Avebury and Silbury Hill are mysteriously spiritual places.

22 of 23

Studley Royal Park Including the Ruins of Fountains Abbey

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Fountains Abbey and Studley Royal Water Garden together make up one of North Yorkshire's most rewarding visitor attractions. Developed over 800 years, it includes a nearly 900-year-old Cistercian abbey—Britain's largest monastic ruin; an 18th century landscaped garden created by a gifted amateur in the era of such celebrity gardeners as Capability Brown and John Vanbrugh; a Jacobean hall and a Victorian Church.

23 of 23

The Tower of London

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William the Conqueror followed his conquest of Britain with a frenzy of castle building. The White Tower, at the center of the fortress now known as the Tower of London, was started almost immediately, in 1066. With it, William the Conqueror baldly demonstrated Norman power and created a fortress and gateway to London at a strategic bend in the River Thames.

Today the tower remains a military establishment. It also houses the British Crown Jewels, the Royal Armoury and the world's oldest continuous public exhibitions; The Line of Kings, the world's longest running visitor attraction, opened in 1652. Its display of English Kings in full suits of armor besides full-sized wooden horses was originally created for King Charles II after the Restoration of the Monarchy. It has been on continuous exhibition and popular ever since.

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