#Solution

[classycookiexo]

True, the more energy we give to the problem, the bigger it seems. Shifting focus to solutions brings peace 💗

[picsfortheday]

[a-path-by-the-moon]

[roakkaliha]

watch the moon

[merpmonde]

Sangaku Saturday #4

In the previous info post, we went over the debate on the religious aspect of sangaku, and the fact that the absence of prayers on these tablets was more puzzling to some than the mathematics. As such, the tablets are not ema prayer tablets, but donations, which usually don't feature prayers on them. Case in point, some consecrated sake and French wine seen at Meiji-jingû in 2016.

Beyond wishing for good fortune and health, such donations serve two very worldly purposes: to contribute to the life and prestige of the shrine or temple (having a famous contributor makes the shrine famous by association), and to advertise the donor in return, as their name is on display. See this large torii at Fushimi Inari Taisha paid for by TV Asahi (テレビ朝日).

With that in mind, Meijizen's cynical comment from 1673 that sangaku aim "to celebrate the mathematical genius of their authors" may not far from the truth. The authors of sangaku are looking to gain notoriety through the publicity that the shrine or temple provides. But was the bemused Meijizen the target audience?

More on that in a couple of weeks. Below the cut is the solution to last week's problem.

The solution to the first problem (below the cut in this post) is the key. Name K, L and M the intersections of the three circles with the horizontal line. Then, by using that previous result,

Indeed, as in that problem, we can construct three right triangles, ABH, ACI and BCJ and apply Pythagoras's theorem in each.

Now, it suffices to note that KL = KM + LM, so

or, dividing by 2*squareroot(pqr), we get the desired result:

Inverting and squaring this yields the formula for r:

This gives us the means to construct this figure on paper using a compass and a marked ruler. Having chosen two radii p and q and constructed the two large circles (remember that AB=p+q) and a line tangent to both, placing M and C is done after calculating the lengths IK=CM=r and IC=KM=2*sqrt(pr).

[whatareyoureallyafraidof]

[cyberianpunks]

[motocrunch]

Skip the work and go riding :)

[bretzkysbs]

"How my girlfriend "handled" a spider, and is too scared to clean it up"

[lifenconcepts]

THE ANSWER YOUVE BEEN SEARCHING FOR!

I think it’s dumb that people expect others to constantly put on a facade of their choosing, none of us are ever stationary and thats why you should never take anything said to you in the heat of the moment to heart - hell, that’s why you never take insults from strangers to heart! You never know what someone is going through in their life currently, or even just lately. It’s illogical to come to sudden conclusions by hearing a single phrase, or asking one question - there are billions of explanations and various variants of how one might be feeling, why, and how that translates into their actions or words. Some can’t understand why they act the way they do, some can turn it to words, and some are set on a single definition that they’ve long outgrown, the human mind will never be as direct as we want it to, and conditioning will only lead to a deep mask imbedded in everyone and imposter syndrome and alike. See your own self for a second then try to imagine every single person you’ve met to bare such a depth, maybe they don’t have the same interests or stress as you but they just as likely have dreams, worries, strengths, goals, weaknesses, ideas, and lingering emotions and interactions that weigh on their consciousness, hell - even good moments can still loom like ghosts of our past. Nearly all have some sort of tint to the world, to some - it’s full of opportunities and new experiences, to another - it’s full of horror and pain. And all can be right in their own way but the little bubble we live in often has its own flavour, some things it lets in to reassure our prior beliefs and reflecting things that go against it, nice if you want to live a real life fairytale, and a destructive game if you have a self-destructive set of ideologies and core beliefs. Change can happen, but, it’s up to you wether that change is good. To some, social connection is a key component of their lives and every day success, to others, simply being around other people is like being suffocated by a brick wall. Never judge another for not doing what you personally consider as “right”, the goal of spirituality is simply to recognise that there can be a certain base to aspire to and steps taken forth to get to that level, but the goal isn’t to convert others to some set rule and use hope to indoctrinate but to achieve what you set out to do and see results. The main purpose of life is simply to find your own happiness and path of what to do, achieve fulfilment, take care of yourself, and be kind to others even if you don’t exactly like them. This doesn’t mean spend every day working you ass off to one day relax as much as you want, nor allow yourself to be a pushover and fail to stand your ground. It simply means to find what works for YOU and specifically you. Find what makes you feel loved and satisfied and incorporate it as much as possible within your existence. You don’t need to go searching for this uniform answer that helps others and will of course help you, because not everyone benefits from yoga, not all are able to grasp meditation, and journaling and the sorts can be an alien concept to some. This doesn’t mean give up at first step, rather, if you tried and it just isn’t working at all, then don’t force it. It’s honourable to fight for yourself but if you spend more time into wanting something to help you rather than actually helping your self, then maybe it just isn’t meant to be. Remember, no matter what you do - there will exist people who love you and people who hate you, and so, you must only focus to make your own being within your skin comfortable.

[classycookiexo]

[eugenedebs1920]

Yes yes… We’ve all been told ‘this is the most important election blah blah’. Yet, it’s definitely up there 😬 Scientists created a grouping of planetary thresholds in order to sustain life on Earth. It is 6 out of the 9 thresholds have been breached. The ocean soaks up so much of our waste & is reaching a tipping point. The article is real informative but shocking. This year it shouldn’t be about taxes and “kitchen table” politics. This election is about the future. This election is about SCOTUS allowing the poisoning of our planet, and frankly our society. It’s about moving forward with remedying decades of neglect & polluting. KamalaHarris will address these issues. Do you think DonaldTrump, a man who thins climate change is fake, will do the same?

https://apple.news/A5xFzsV5MSsuHnvc8tAC9CA

[a-path-by-the-moon]

[roakkaliha]

been feeling artblocked, so heres baby solution + wolfstar

[merpmonde]

Sangaku Sunday #6

We are about to solve our first sangaku problem, as seen on the tablet shown above from Miminashi-yamaguchi-jinja in Kashihara.

First, we should conclude our discussion: what are sangaku for? There's the religious function, as an offering, and this offering was put on display for all to see, though not all fully understood the problems and their solutions. But a few people would understand, and these would have been the mathematicians of the time. When they visited a new town, they would typically stop at a temple or shrine for some prayers, and they would see the sangaku, a sample of what the local mathematicians were capable of. Whether the problems were solved or open, the visitor could take up the challenges and find the authors to discuss.

And this is where everything lined up: the local school of mathematics would have someone new to talk to, possibly to impress or be impressed by, and maybe even recruit. With the Japanese-style mathematics of the time, called wasan, being considered something of an art form, there would be masters and apprentices, and the sangaku was therefore a means to perpetuate the art.

Now, what about that configuration of circles, second from right on the tablet?

Recall that we had a formula for the radii of three circles which are pairwise tangent and all tangent to the same line. Calling the radii p, q, r, s and t for the circles of centres A, B, C, D and E respectively, we have

for the circles with centres A, B and C (our previous problem), and adapting this formula to two other systems of three circles, we get

for the circles with centres A, C and D, and

for the circles with centres B, C and E. Add these together, and use the first relation on the right-hand side, we get a rather elegant relation between all five radii:

Of course, we can get formulas for s and t,

r having been calculated previously using just p and q, which were our starting radii.

For example, setting p=4 and q=3, we get, approximately, r=0.86, s=0.4 and t=0.37 (this is the configuration shown in the figure, not necessarily the one on the tablet - I will be able to make remarks about that on another example).