#tetrahedron net

[art-of-mathematics]

2. Fold an equilateral triangle grid /tetrahedron net

2.1.) Starting with the 120 degree angle we continue to fold it in half to get a 60 degree angle:

2.2.) Then fold it this way:

(The right (depicted in orange) edge of the paper needs to lay on top of the top side of the paper. )

Repeat these steps until you have only a stacked triangle.

2.3.) Unfold it:

2.4.) Now valley-fold all folds - all folds have to be in the same direction.

This results in a line of almost 5 entire equilateral triangles. - For a tetrahedron you need more than 4 triangles. (4 equilateral triangles aligned in a line is one of the many variations of a tetrahedron net.) (A tetrahedron has 4 faces. But the 5th triangle is needed to insert as a flap - and make the shape sturdier. )

2.5) Assemble:

Roll it up until the 5th triangle stays up as a flap.

Then insert the flap into the face:

Finished:

I hope you enjoy my lil shape and shape tutorial. 🌈

- - -- ---

DIN Ax (1:SQRT2) paper might work well for that.

With square (1:1) paper it does not work.

Quick friend-shapes needed to be made.

(meaning: I was bored and needed to do something with my hands and brain.)

[knotty-et-al]

Colorful nets of Truncated Tetrahedron and Tetrahedron

[dailypolyhedra]

Polyhedron of the Day #63: Triakis tetrahedron

The triakis tetrahedron is a Catalan solid. It has 12 faces, 18 edges, and 8 vertices. It is also known as the kistetrahedron, as it is the Kleetope of the tetrahedron (i.e., this shape can be obtained by attaching triangular pyramids to each face of a tetrahedron). Its structure is similar to that of the net of the four-dimensional 5-cell. Its faces are all isosceles triangles. Its dual polyhedron is the truncated tetrahedron.

Triakis tetrahedron GIF and image created by Cyp using POV-Ray, distributed under a CC BY-SA 3.0 license.

[lady-inkyrius]

@kaiasky

Yeah, tetrahedral maps are great! I think it's mainly because when the map is centred on the north pole you sort of get three "prongs" of land extending from the centre (South America, Africa, and Australasia/Oceania), which then obviously fit into the three points of the Lee Tetrahedron's resulting triangle really well.

This manages to keep most of the land away from the distortion points at the centre of each side and at the corners, with the noticeable exceptions of the area around Myanmar & Bangladesh, Hawaii, and Antarctica.

I'll put the rest of this under a cut because it got quite long.

There are a couple of different ways to fix this, centring it on 65°N 30°E rather than the pole manages to put Antarctica back in one piece and now the only noticeable distortion is that Sri Lanka and southern India are way too big, (There are probably better options than this for the centre but I couldn't find them)

Of course, you might also notice that the three "prongs" are really closer to 90° apart than 120°, which is where the popularity of octahedral projections like Cahill's and Waterman's projections probably comes from, but I quite like the Peirce Quincuncial for this, and if you take the top octant and split and rotate it round, you get a layout very similar to the tetrahedron. The most noticeable distortion here is probably Papua New Guinea being slightly to big.

(And like in this post from last week, you can also move the bit of Antarctica at the bottom to get this)

A lot of people have tried to use Lee's Tetrahedron as a base and either rearrange it into a rectangle (Markley and CALM) or a half hexagon (Lee-Concialdi and Lee-Xarax), it's just a really good projection if you need something conformal with low area distortion.

The only oblique aspect rearrangement of the Peirce Quincuncial I've seen is Grieger's Triptychal (see the Markley and CALM link).

A polyhedral projection I really want to see is one based off a rhombic dodecahedron, a polyhedron with twelve rhombi for sides.

It's net isn't particularly pretty because the rhombi it uses don't tesselate, however this paper does a really interesting thing by compressing the net to squares rather than rhombi.

It's original use case is as replacement for cubemaps which can cause a loss of visual quality where the distortion is large, but I want to see what it would look like for a world map.

The projection they're using for each face (gnomonic I think) isn't conformal so you get discontinuities between face, but you might be able to change it so it is, sort of like how the Peirce Quincuncial uses right angle triangles for each octant rather than equilateral triangles like the Cahill while still maintaining conformality, but I don't know how I'd work out the maths for that, iirc Peirce did it with complex analysis.

[map-projection-showdown]

DYMAXION vs LEE TETRAHEDRON

Dymaxion Polyhedral Compromise Round 1: [Dymaxion vs Stab-Werner]

Lee Conformal Tetrahedron Polyhedral Conformal Round 1: [Lee Tetrahedron vs Equirectangular]

And the battle of the polyhedral projections is here!

Despite lacking in dynamicity, maximality, or tension, the Dymaxion has become very popular due to its low area and angle distortion, as well as its unique shape. The icosahedral net allows it to represent all the continents without any breaks, and as such has been used for things like maps of human migration.

Lee's tetrahedron uses a smaller platonic solid, which results in a much cleaner net, though does come with the downside that Antarctica is split in three in the north polar aspect. As it's represented as a triangle this projection can tesselate a plane, which means that its easy to move parts around to rearrange it into a better shape, such as CALM, a version of Lee rearranged into a conformal version of the Authagraph projection, or the Lee-Concialdi which is also in this tournament.

[Direct comparison on map-projections.net]

[link to all polls]

[paulfc]

Tetrahedron pull up and pull down net

I should really have done this one first, since it is the first of the Platonic solids, having four faces.

I mentioned previously about the hinges between the faces being sticky tape and it turns out if you put the faces too close together, as it fold up then there is tension created in the hinge which tries to pull it flat or at least lift it a bit.

Seems to pull up and pull down quite nicely though.  I suppose it’s the octahedron next.

[xingchen-official]

Like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. It has two such nets.

[myxotesta]

Vanaard

"A gigantic net, woven from thick steel cables. Its meshes were so wide that entire ships could fit through them. It stood dome-shaped over the outlines of an impressive city. Triangular prismatic skyscrapers with numerous tourelles and residential towers filled every available nook and cranny beneath the protective net. Curving structures sprang from orange streets and flowed into flower-shaped glass crowns where new planes landed steadily. The interwoven web of houses under the mesh dome reminded them of a brain. Above the flesh-colored tangle hovered a massive platform holding a giant tetrahedron: the flying island Eleva."

My Fantasy / Scifi novel "Gründämmerung" will be available soon in English and German.

[harrelltut]

iGenetically [IG] MONETIZED [I’M] Me [ME = U.S. Michael Harrell = TUT = JAH] on Earth [JE = JESUS]… from 1968 [VI]… since iMathematically [I’M] MULTIPLIED [I’M] Me [ME = U.S. Michael Harrell = TUT = JAH] into Highly Complex [ADVANCED] Ancient Interactive Ægyptian METAPHYSICS that Algebraically Calculate [MAC] Abstract Numerical [MAN] Expressions [ME = U.S. Michael Harrell = TUT = JAH] Naturally ENCRYPTING TECHNICAL [NET] COMPUTER Messages [CRYPTOGRAPHS]… QUANTUM HARRELL TECH SHADOW MILITARY.gov [NWO]... Certifiably AUTHENTICATE with Technical [CAT = SPHINX] DNA RECORDS of Biblically Ancient [BABYLONIAN] SUN_KING [SUNKEN] DOME Continent [D.C.] ROYALS from NEBUCHADNEZZAR's Astronomical [RNA] SUN DYNASTY of Highly Complex [ADVANCED] HURRIAN INSCRIPTIONS [HI = HITTITES] of Ancient [HA = HARRELL] Earth TETRAHEDRON [E.T.] SYMBOLS Esoterically & Artistically [SEA] COMMUNICATING… Ægyptian RAMESES’ Lost American [L.A. = NEW Atlantean] SOVEREIGN [ROYAL] MAYAN FAMILY CODES that Ancestrally TELECOMMUNICATE [CAT = SPHINX] GENERATIONAL AZTEC [GA = ROMAN] RELIGION SECRETS from Our Primordially ANCIENT [PA = SUPERNATURAL] COSMOLOGICAL Human Junk DNA/RNA of Ancient [RA] FUTURISTIC Astronomically Intelligent ANUNNAKI [AIA = AMÚN] SUN GOD [RA] MANIFESTATION [RAMESES] POWERS… Immaculately MATERIALIZING [I’M] DEEP [I.D.] IN:side My New [NU] ATLANTEAN & LEMURIAN SEA Continent Altitude of PANGÆA [CAP = CAPRICORN] from 2012 [V]... in 2021 [V]… since iScientifically Envisioned Afterlife [iSEA] Human SPIRIT WORLDS from My Ancient OLMEC STAR FUTURE of ORION’S 9th SOUL Dimension of 5000 https://www.instagram.com/p/CRcvO1clxi3/?utm_medium=tumblr

[art-of-mathematics]

I made this collage many years ago - called it "Theory of Everything".

Infodumping:

The guy in the middle is Edward Frenkel (I literally forgot his full name, I knew he was called Edward Fra/enk... and then I couldn't remember... ) but he's one of my idols. Plus he wrote about the Langland's program, which is like the unified/ theory of everything in the math field. Also he is a human of value, as I have read in his autobiography "Love and Math" (which I sadly never finished, because so much thoughts and literally too few time to analyze everything til the end.)

The green thingy in the top right is a Lorentz attractor. (Sidenote: I have huge issues with left and right distinction, because my brain also thinks "reversely"/ mirrored, plus my dissociative mind tends to see me from above and what the fuck, do you mean this left or the left I see when I rotate my view by 180 degrees?)

The entire piece consists of chaos nd portrays my mind. (Secret order is hidden.) Intuitive paths...

The main parts are tesseract-nets, 3-dimensional nets of a four-dimensional cube, or hypercube/tesseract.

Quantum foam included! This word was introduced by John A. Wheeler as he tried to describe the immense activities on the quantum scale. Cool thing is: it, say rhe spacetime net, *appears* smooth at the first look, but if you zoom into it, it will look like "foam", chaos intensifies!

This brings me to the concept that all pure deterministic values (1 OR 0) are a product of a superordinated probabilistic system as proposed in Quantum mechanics (procentual probabilities, like 1 AND 0). A probabilistic system gets deterministic if observed/interacted with. Leads to wave function collapse. The wave function collapse is a prduct of a cross-section of the wave function, so to speak.

This leads to "Euler's identity: a formula that describes that cosine and sine are just 2-dimensional pictures of the 3-dimensional helix.

Also: what came first: Natural, imaginary or real numbers?

My solution for a theory of everything: Trying to make General Relativity fit with Quantum Mechanucs on the SAME LEVEL is line comparing apples with apple trees... you see what I mean: one thing is sub-ordinated, the other superordinated, one thing is nested onto the other...

Leads to ideas regarding recursion and furthermore self-reference combined with thoughts bout Gödel's Incompleteness.. (That's gives it a whole new level...*pun intended*)

Fractal pattern, fractal dimensions and dimensions in general. A dimension is just how much dots (0-D) are connected: 1D: the line (1 dot has one connection to another dot), 2D: the square (one dot has 2 connections to another dots), 3D: the cube (one dot has 3 connections); 4D: the hypercube (one dot has 4 connections to other dots)... yoz see a pattern?

Leads indirectly to quaternions and other hypercomplex number systems...

Relation to Platonic solids? Paterns of three, (the tetrahedron, octahedron and icosahedron), four (cube) and five (dodecahedron)

Data science/ statistics side thought - everything is just mathematical information. Physical reality is embedded in Mathematical reality, and is the rounded version of the infinitely complex Mathematical reality. Loops to the thought of wave function. Something is just *real* if it interacts, otherwise it's everywhere and nowhere simultaneously. Hence if it interacts it needs to be "rounded" aka the probabilities need to simiplify themselves.

Side note to Synergetics (After Buckminster Fuller after whom the C60-molecule "Bucky-ball" is named: regarded as geometry of thinking; and after Hermann Haken, (who is still alive and almost 100yo and lives in Stuttgart currently and was born in Leipzig, just 150km away from where I live.): acoording to Haken, Synergetics is the study of interaction. One important upshot: microscopic chaos creates macroscopic order. ...backloop to my idea about deterministic and probabilistic systems (equivalent)

Utter amazement of this terribly chaotic stream of thoughts... that's when you think recursively AND self-referently... i am just imitating the logic of the universe... a living paradox whose only escape is the meta level (reference to implications of gödel's incompleteness theorems)...

Whatever. I have head aches now.

That was intense brain storming...

And keep in mind:

Speculation and skepticism are both the key to theoretical discoveries.

(Inconsistent ideas get transformed/ erased.)

This thought stream is literally the upshot of my now already 9 years old hypothesis, or some few key concepts...

[kitwallace]

Paths on Polyhedra

Bob Bosch posted a picture of a rather lovely wooden object.  A grove had been machined around a cube in which a steel ball was free to run.  

Ever willing to steal other folk’s great ideas, I set about implementing the object in openSCAD.  My first attempt was a hack.  I made a quarter-circle tube out of hulled spheres and translated and rotated copies to lie with the right orientation on the sides of a cube. The tubes were then removed from the cube using difference(), leaving a continuous grove. Displacing the tube inwards created a lip to capture the ball bearing.

It was only when I had made the construction that I realised that the path was closed and touched every face so it was a Hamiltonian circuit around the cube faces. [I noted later that it was the waist of the cube too]

A better construction would take a list of face indexes and convert to the sequence of quarter tubes.  Given the definition of the polyhedron as a set of faces and vertices, it should be possible to construct the tubes in their position on the surface. 

It helps to number the faces:

In openSCAD you can rotate this object to find a cycle of  [0,1,4,3,5,2] .  Each tube runs across a face from mid-point of one edge to mid-point of the next. 

The construction interpolates along the arc to to create a path as a sequence of points, then places spheres on the path which are hulled together. Now any path can be constructed, like the loops around the bare vertices:

This code is generalised so we can construct paths on other polyhedra:

Tetrahedron:

Octahedron: 

When edges are adjacent, the path is an arc.  If they are not, the path is a straight line between the edge midpoints.  This is also a circuit, but less symmetrical :

Performance is slow and an alternative construction using a constructed polyhedral net would be faster and allow a tube profile which could also round the sharp edge.

It would be fun to commission one of these objects - with a captive ball - in metal from Shapeways.  I fancy making a dodecahedral version, using the vertex Hamiltonian of its dual icosahedron. I think its rather curious that Hamilton’s Icosian game was played on the vertices of a dodecahedron, rather than on the faces of a icosahedron, which would seem to be easier to draw. 

 Code on Github.  

[dailypolyhedra]

Polyhedron of the Day #19: Regular tetrahedron

The regular tetrahedron is a Platonic solid as well as a deltahedron. It has 4 faces, 6 edges, and 4 vertices. It can also be referred to as a 3-simplex, as it is a specific 3-dimensional case of the general n-dimensional simplex, or a triangular pyramid. It has 2 possible nets. The regular tetrahedron is self-dual.

Tetrahedron GIF created by Cyp, distributed under a CC BY-SA 3.0 license.

Tetrahedron image created by Kjell André, distributed under a CC BY-SA 3.0 license.

[quadrantvideogames]

The original donkey kong had four stages. Later Donkey kong games still included these four stages or at least one of them. The killscreen is at the 22nd level because it reaches the 256 limit. 256 is four to the fourth power.

http://errors.wikia.com/wiki/DK_kill_screen

https://en.wikipedia.org/wiki/Donkey_Kong_(video_game)

Legend of Zelda, a Link to the Past and Four Swords has four Zeldas and four Great fairies. Zelda games are permeated with fours. There is even a Zelda tetraforce theory that argues that the triforce is in fact the tetraforce. In Four Swords Adventures, four Links are always at play.

https://en.wikipedia.org/wiki/The_Legend_of_Zelda:_Four_Swords_Adventures

https://en.wikipedia.org/wiki/The_Legend_of_Zelda:_A_Link_to_the_Past_and_Four_Swords

In Zelda Oracle of the Seasons the environment shifts between the Four Seasons, and there is a lot of repetition of four in the game.

https://en.wikipedia.org/wiki/The_Legend_of_Zelda:_Oracle_of_Seasons_and_Oracle_of_Ages

https://en.wikipedia.org/wiki/Xbox

There are four action buttons in the X box. The x in the X box controller is the quadrant.

A

X

B

Y

The X in Direct X is the quadrant.

https://en.wikipedia.org/wiki/DirectX

Left 4 Dead is a popular video game

William "Bill" Overbeck (voiced by Jim French), a Vietnam veteran; Zoey (voiced by Jen Taylor), a college student; Louis (voiced by Earl Alexander), an IT analyst, and Francis (voiced by Vince Valenzuela), an outlaw biker—make their way through the city of Fairfield, only to discover that the infection is creating more dangerous mutations

https://en.wikipedia.org/wiki/Left_4_Dead

https://www.mariowiki.com/Mario_Kart_64

There are four cups in Mario Kart 64

Mushroom cup

Star Cup

Flower cup

Special Cup

http://mariokart.wikia.com/wiki/Mario_Kart:_Double_Dash!!

There are 16 tracks in Mario Kart Dash. There is 16 squares in the quadrant model.

https://en.wikipedia.org/wiki/Mario_Kart

Each Mario Kart game has 16 courses in four Cups. Therefore Mario Kart is a four by four quadrant model.

https://en.wikipedia.org/wiki/Mario_Kart:_Double_Dash

Mario Kart Dash has 16 courses divided into four tracks. It also allows 16 players. There are 16 squares in the quadrant model.

The four Bosses in Street Fighter II are shown below.

http://archive.boston.com/ae/movies/articles/2011/10/16/multiplied_by_four/?page=2

https://en.wikipedia.org/wiki/Mario_Kart_DS

Each cup in Mario Kart DS has four levels too. But there are eight cups. So there is 32 levels in all.

Sagat

Balrog

Vega

M. Bison

There are four elements in a 4X video game

https://en.wikipedia.org/wiki/4X

Explore

Exploit

Expand

Exterminate

The Bartle taxonomy of video game players elucidates four types

https://en.wikipedia.org/wiki/Bartle_taxonomy_of_player_types

Jon Radoff has a four quadrant model of video game players

immersion

achievement

cooperation

competition

https://www.slideshare.net/gzicherm/jon-radoff-designing-for-user-motivation-understanding-the-four-quadrants-how-they-affect-your-product-design

Tetris is one of the most popular video games of all time. Tetra means four. The game pieces are made up of four segments

https://en.wikipedia.org/wiki/Tetris

https://en.wikipedia.org/wiki/Tetromino

https://en.wikipedia.org/wiki/PlayStation_Controller

The playstation controller has four action buttons in a quadrant formation. They are

green triangle

a blue cross

a red circle

pink square

One of the buttons is a cross.

In the popular video game Borderlands there are four main characters

https://en.wikipedia.org/wiki/Borderlands_(video_game)

Roland the Soldier

Lilith the Siren

Mordecai the Hunter

Brick the Berserker

The tetractys appears in Assassin's Creed. The gematria of the tetragrammaton tetractys is 72. The number 72 plays a significant role in Assassin's Creed

http://assassinscreed.wikia.com/wiki/Pythagoras

A tetractys of the letters of the Tetragrammaton adds up to 72 by gematria.

The 72 names of God, according to Jewish mystics, are divided into four columns. The four columns are represented by the four letters of the tetragrammaton. Again, other numbers appear, but the four is still dominant.

http://ataribreakout.org/atari-breakout-game

Atari breakout is a classic video game. In the game there were four colors of blocks. Each color of block would deflect the ball at a different speed.

In 2010 taco bell had a rerelease of Breakout. Four classic atari games were sold.

https://en.wikipedia.org/wiki/Breakout_(video_game)

http://www.zeldainformer.com/the_entirely_valid_completely_believable_theory_of_a_tetraforce/

In Zelda there is a theory called the tetraforce theory.

The triforce is made up of three yellow triangles. The three triangles represent wisdom, power, and courage. The Hylian Shield in Ocarina of Time, however, has a fourth triangle below the triforce symbol. THe shield is illustrated above. It was hypothesized that this fourth triangle was the fourth force in the triforce. We see, thereby, the quaternity principal at play. The three is becoming four, or at least there is a different fourth component that does not seem to belong.

The number four is significant in Zelda. The prominence of the number four in Zelda leads “tetraforce conspirators” to believe that there is a fourth part of the triforce.

Some examples of four in Zelda are

There are four giants in Majora's Mask.

https://zelda.gamepedia.com/File:Four_Giants.jpg

https://zelda.gamepedia.com/The_Legend_of_Zelda:_Four_Swords

In The Legend of Zelda: Four Swords, there is a legendary sword that can turn a person into four persons. Link in the game, uses the magical sword and becomes four persons.

The light spirits in Zelda are four God like entities.

http://zelda.wikia.com/wiki/Light_Spirits

Ordona

Eldin

Faron

Lanayru

In Zelda there are four mirror shards, but only three need to be retrieved by Link.

http://zelda.wikia.com/wiki/Mirror_of_Twilight

The mirror shards are in four locations

The Mirror Chamber atop the Arbiter's Grounds

The Temple of Time in the Sacred Grove

The Snowpeak Ruins at the summit of Snowpeak

The City in the Sky located in the heavens

In the game Zelda, Zelda is Tetra. Tetra is also called Dazel in the game. Dazel is an anagram of Zelda. Tetra means four.

Artwork of Tetra from The Wind Waker

“Tetra may also be a pun of the word Tetrahedron as the symbol of the Triforce is the net of a tetrahedron.”

A swastika shaped third dungeon in Zelda.

The Book of Magic in Zelda has a cross on it.

http://zelda.wikia.com/wiki/Book_of_Magic

http://zelda.wikia.com/wiki/Hyrule_Warriors

In Hyrule Warriors, a legendary figure defeats an evil being and splits his soul into four pieces. Three of the pieces are scattered across space and time, but the fourth piece is held at the Temple of the Sacred Sword.

http://zelda.wikia.com/wiki/Cross

The cross is a quest item in Zelda II the Adventures of Link.

There is a game called Dance Dance Revolution X. X is a quadrant.

https://en.wikipedia.org/wiki/Dance_Dance_Revolution_X

Dance Dance revolution dance platforms have four pads pointing to the four directions.

https://en.wikipedia.org/wiki/Dance_Dance_Revolution

Dance pads can be used with a controller for dance dance revolution, and soft, portable dance pads can be used at home.

https://en.wikipedia.org/wiki/Dance_pad

Gauntlet was one of the first arcade games where you could play more than two players at once. Gauntlet allowed for four players to play at one time. The four players are

Questor, an Elf

Thor, a Warrior

Thyra, a Valkyrie

Merlin, a Wizard

Pac Man was one of the most popular video games/arcade games of all time. In the game Pac Man must run from four multi colored ghosts. The four ghosts fit the quadrant pattern, in that the fourth ghost is different. The first three ghost’s names rhyme, but not the fourth’s.  Also, at the beginning of each level, three of the ghosts are together in a rectangular box, but the fourth is outside.

https://en.wikipedia.org/wiki/Pac-Man

The initial configuration of Google's Pac-Man banner is shown above

There are four different mazes that appear in different color schemes in Ms. Pac Man. They are

https://en.wikipedia.org/wiki/Ms._Pac-Man

The pink maze appears in levels 1 & 2

the brown maze appears in levels 6 through 9

the light blue maze appears in levels 3, 4, & 5

and the dark blue maze appears in levels 10 through 14. After level 14, the maze configurations alternate every 4th level.

Three of the four mazes have two sets of warp tunnels (four warp tunnels). The fourth is different.

The Mega Drive/Genesis, Master System, and NES versions, by Tengen, and the Super NES version, by Williams Electronics, used four sets of mazes for Ms. Pacman. They were

the original arcade mazes

smaller mazes

bigger mazes

"strange" mazes

https://en.wikipedia.org/wiki/List_of_Star_Fox_characters

In star fox assault there are four members of the Star Fox team. Star Fox is one of the most popular video games of all time

https://en.wikipedia.org/wiki/List_of_Star_Fox_characters

, Slippy Toad

Falco Lombardi

Fox McCloud

Krystal

https://en.wikipedia.org/wiki/Pokémon_X_and_Y

Pokemon X is a role playing video game. The X is the quadrant.

The game space invaders is infamous for its musical four note loop.

https://en.wikipedia.org/wiki/Space_Invaders

There are four base shelters in the original space invaders video game.

http://www.classicgaming.cc/classics/space-invaders/play-guide

There are also four types of villains that you have to destroy. There is the small invaders, the medium invaders, the large invaders, and there is the transcendent fourth mystery ship/ufo.

https://en.wikipedia.org/wiki/Classical_elements_in_popular_culture

Atari’s Swordquest featured four video games. The four video games were titled under the four elements.

SwordQuest: EarthWorld

SwordQuest: WaterWorld

SwordQuest: FireWorld

SwordQuest: AirWorld

https://1.bp.blogspot.com/-9gARUDTuCGo/WDoihSxdseI/AAAAAAACcXQ/H0WKF_x5ARIafmAP11ZVoqR0GTNw7184gCLcB/s1600/assassins%2Bcreed%2Bmovie%2Bbillboard.jpg

http://www.dailybillboardblog.com/2016/12/assassins-creed-movie-billboards.html

Quadrant

The Assassin's Creed poster shows the assasin in cruciform.

https://en.wikipedia.org/wiki/4X

The term "4X" originates from a 1993 preview of Master of Orion in Computer Gaming World by Alan Emrich, in which he rated the game "XXXX" as a pun on the XXX rating for pornography. The four Xs were an abbreviation for "EXplore, EXpand, EXploit and EXterminate".[1]

Explore means players send scouts across a map to reveal surrounding territories.

Exploit means players gather and use resources in areas they control, and improve the efficiency of that usage.

Expand means players claim new territory by creating new settlements, or sometimes by extending the influence of existing settlements.

Exterminate means attacking and eliminating rival players. Since in some games all territory is eventually claimed, eliminating a rival's presence may be the only way to achieve further expansion.

[map-projection-showdown]

EQUIDISTANT CONIC vs DYMAXION

Equidistant Conic Conic Equidistant Round 1: [Equidistant Conic vs Robinson] Round 2: [Equidistant Conic vs Sinu-Mollweide]

Dymaxion Polyhedral Compromise Round 1: [Dymaxion vs Stab-Werner] Round 2: [Dymaxion vs Lee Conformal Tetrahedron]

We've got projections with very different aims facing off here.

While I'm presenting the Equidistant Conic using a map of the whole world, it is rarely used like this, its most common use case is for mapping regions that are longer east-west than north-south, like Russia or sometimes the US, as standard parallels can be chosen to reduce distortion at specific latitudes.

The Dymaxion on the other hand is specifically designed to view the entire world from a new perspective with no land interruptions, the aspect showing how close continents are around the north pole, and is perhaps one of the more famous of the projections that do this. While the icosahedral net of the Dymaxion does give it extremely low distortion for a compromise map, it does result in its rather... unique shape.

(Fun fact, the equirectangular projection could actually be considered a special case of the equidistant conic, as you get an equirectangular map if the standard parallels you select are symmetrical either side of the equator.)

[Direct comparison on map-projections.net]

[link to all polls]

[paulfc]

Pull up nets, cube, tetrahedron, octahedron, dodecahedron and icosahedron

I was reading a paper about pull up nets and it suggested that not all of the nets of a cube necessarily can be made to pull up.

Challenge accepted, I have made all eleven of the cube nets and depending on your definition of 'pull up' all of them can be made to pull up.  I accept that not all of them do so elegantly but they do pull up into a cube.  Given that there are many ways to loop thread through them improvements may be possible if there were an easy way to determine the most effective route.

When considering this, knowing which bit of the net end up where in the cube, that is to say which corners in the net end up in which vertices in the cube helps in choosing a thread route that pulls these together.

Tacked on to the end are the only two unique nets of the tetrahedron, one of the eleven nets of the octahedron and one each of the thousands of nets of the dodecahedron and the icosahedron.  I may get round to the other ten nets of the octahedron but not the thousands of the dodeca and icosahedrons.  The number of nets increases geometrically with the number of faces and side of the polyhedron I don’t know if there is a formula but it does get very silly very quickly.

These are card with sticky tape hinges and the eagle eyed will have seen that sometimes they flex the wrong way and yes the card came from pizza boxes, I eat a lot of pizza and always save the cardboard for making maquettes.

[ais-nutraceuticals]

Treat sleep disorder with the help of CBD

CBD, short for cannabidiol, is a chemical made up of the Cannabis sativa plant, also known as hemp. CBD is a chemical found in marijuana. CBD doesn't contain tetrahedron (THC), the psychoactive ingredient found in marijuana that produces a high.

CBD for sleep also helps you wind down by addressing some of the root causes of restlessness and stress. For example, many CBD products contain terseness, aromatic compounds with therapeutic benefits. Terseness help decrease stress and anxiety, and some even have sedative effects.

Anxiety is one of the most common causes of insomnia; worrying leads to sleep problems, and having difficulties in sleeping can cause much stress. Research says that low doses of CBD are stimulating while high amounts of CBD are sedating. Discrepancies in inexperience can also be attributed to CBD administration and dose.

Types of CBD extracts

Full-spectrum.Full-spectrum isolate

Broad-spectrum

"Pure" CBD isolate

CBD administration methods include:

oil

sublingual spray

capsules

edibles

vaporized oil or cannabis flowers

How to check the legitimacy of the CBD product-

Be sure that the label contains the following:

Name and the manufacturer or distributor

Cannabinoid content

Ingredients which include non-cannabis additives 

Net quantity

Lot, batch, or control number

Product name 

Expiration date

Directions on how to use and store the product

Suggested serving size or dosing information

Warnings\contradictions

If a CBD product fails to include all of the above information on the label, it may not be legitimate.

Other Benefits of CBD –

It may relieve pain: CBD oil is a popular natural pain relief treatment.

It may help control epilepsy and seizures

It may reduce anxiety and depression

It may assist with cancer-related symptoms

It may help with acne

What Forms Does CBD Come In?

There are several common forms of CBD:

Oral solutions, like the FDA-approved drug Epidiolex

In Oral sprays that are applied under the tongue

In Oils and tinctures, sometimes described as drops or droplets

In Vapes and vape juices to use in a vaping pen

In Edible items, such as gummies, chocolates, or cookies, and beverages, such as coffees and teas

It also comes in Pills and capsules

In also Topical solutions, including lotions, creams, patches, gels, and ointments

 CBD is an increasingly popular substance in Australia. Many people take CBD as part of their nighttime routine to help them find calm in between everyday chaos. It would be advisable to start CBD with a low dosage and gradually increase it over a long period. For example, you may want to first use just a couple of drops of CBD oil and then begin to add more drops each time you use it until you find the amount that works for you.