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gorrus · 3 months

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#hexagon #polygon #octagon #pentagon #heptagon #enneagon #decagon

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notreallyherehahaha · 2 years

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A Tessellation Featuring Regular Enneagons and Triangles, as Well as Concave Octahegons and Six-Pointed Stars

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#enneagon #geometry #Mathematics #nonagon #tessellation

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squareallworthy · 4 months

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Reverse unpopular opinion ask meme: Irregular polygons

Irregular polygons are awesome. You can do so much more with them, in so many areas, but I'm going to limit myself to talking about just two of them: tiling and triangle centers.

(Did you think I was going to be all snobby toward my irregular friends and give them only grudging approval? Heck no, I love those guys! And so by the rules of the meme I get to infodump about the things I love, so this may be long but you asked for it.)

Let's first talk about covering the plane with copies of a single shape -- a monohedral tiling. And for now, let's restrict ourselves to periodic tilings. All triangles and all non-self-intersecting quadrilaterals tile the plane periodically, so that's not very interesting. All you have to do is place one polygon and then make copies by rotating 180 degrees around the midpoints of the sides.

With five sides, things become more complicated, because regular pentagons don't tile by themselves, but there are fifteen ways an irregular pentagon can periodically tile the plane. Here are four of them that were discovered in 1976 and 1977 by Marjorie Rice, an amateur mathematician.

There are three types of monohedral periodic convex hexagonal tilings.

For polygons with seven or more sides, there are no monohedral periodic tilings using a convex prototile , but there are periodic tilings for nonconvex polygons of any size. Some of them are quite famous.

Most of Escher's work in his Regular Division of the Plane series uses shapes with curves as well as straight sides, so they don't show polygon tilings, strictly, but the patterns do point toward complex tilings that are visually pleasing.

Irregulars can tile aperiodically, too. Here's a pentagon tiling with 6-fold rotational symmetry. It can be extended infinitely, and tilings can be constructed with pentagons for n-fold symmetry of any n>2.

Going back to non-convex shapes, here's the Voderberg tile, an enneagon that forms a spiral tiling. Notably, one copy of the shape can be completely surrounded by two others.

And of course I can't go without mentioning the tiling news of the century: "Tile (1,1)", aka the Hat, aka the T-shirt, a tridecagon (and polykite) that can tile the plane but only aperiodically. IDK if you follow polygon news but this was huge.

Okay, enough about tilings. With tilngs it's pretty easy to get what's going on just by looking at them, but my next topic, triangle centers, requires a bit more explanation. Also there's a bit of jargon, but I will try to keep it simple.

Take an arbitrary triangle ABC. Where is its center? One way you might define it is to find the midpoint of each side and draw a line to it from the opposite vertex. Each line divides the triangle in half, and these three lines (the medians) all cross at a point, the centroid. This works for any triangle, no matter its shape. The point marks the center of gravity of the area of the triangle, and also the center of gravity of its vertices. Based on that, you could consider this the center of the triangle.

Or you could work with angles instead of sides. Draw lines from each vertex that divide the angles in half (the angle bisectors). These all meet at a point called the incenter, which marks the center of the largest circle that fits inside the triangle. To put it another way, it's the point that is equidistant from all three sides. That's another point you could call the center of the triangle.

Or, how about a circle around the triangle instead? From the midpoints of the sides, draw the perpendicular bisectors. Again, they all intersect at a point, the circumcenter, which is the center of the circle that passes through the vertices -- the point that is equidistant from all three of them. So you could also call that the center of the triangle.

Or how about drawing perpendicular lines from the sides again, but having them pass through the opposite vertices (the altitudes)? They coincide at a point called the orthocenter. Isn't that neat? Yet another point we could call the center of the triangle.

But wait -- can we? For an obtuse triangle, the circumcenter and the orthocenter are going to lie outside the triangle. (For the orthocenter of an obtuse triangle, you have to extend each side into a line, and draw the altitude as a perpendicular to that.) Being outside a thing is really not what we have in mind when we talk about the center of the thing. Should we care about that?

Maybe not. Check this out. We'll go back to the circumcircle, and draw tangents to it at the three vertices. The three tangents form the tangential triangle (in blue), which we'll call A'B'C', where A' is opposite A, and so on with B and C. Now draw the circle that passes through A, A', and the circumcenter, and do the analogous construction for B and C (in red). The three circles coincide in two places: the circumcenter and another point called the far-out point. And as the name suggests, this is usually well outside the triangle, even for acute triangles.

There's no reasonable way to call this point the center of the circle. But so what? I just love the fact that the three circles line up like that. I no longer care about finding "the" center of the triangle. I no longer care that "center" is rather a misnomer for many of these points. I just think it's neat that you can draw these constructions on ordinary aysmmetrical triangles and they keep all converging on one point. Want more? Reflect the medians across the angle bisectors, and they all meet at the symmedian point. Or connect the vertices of the tangential triangle with the intersections of the medians and the circumcircle. Those lines meet at the Exeter point. Or, from each vertex, draw the line that splits the perimeter of the triangle in half. These are called the splitters, and they meet at the Nagel point. And on and on and on.

You can simply wander around a triangle, connecting things that relate somehow to vertex A, then do the equivalent thing for B and C, and stumble upon new centers. And there are tens of thousands of these things, constructed with straightedge and compass or by other methods. And there are so many ways to enjoy these things. You can page through the enormous collection and get a kind of stamp-collecting satisfaction just looking at their variety and knowing that they exist. Or you can appreciate the proofs that show that the constructions really do specify a unique point. Or proofs that show that a point constructed to have one property has a surprisingly different property. Or you can notice that the points fall into certain families and appreciate the connections between them. (For instance, the centroid, circumcenter, orthocenter, far-out point, and Exeter point, among others, all happen to lie on the same line, the Euler line.) Or you can convert the points to trilinear coordinates, manipulate them algebraically, and get to know them that way.

But to appreciate them at all, you need to work with irregular triangles. Because here's the thing: in an equilateral triangle, all these points collapse to the same point. Everything simplifies to a single center, and the incredible wealth of invisible structure that teems inside every ordinary triangle is gone.

You have finally found "the" center, but at what cost? Symmetry is death. Only through asymmetry will the vast truth of the triangle be revealed to you.

And those are just a few of the reasons irregular polygons are cool!

#polygon representation

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revenant-coining · 2 years

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Digonlexic, Enneagonlexic, Gigagonlexic, and Hectogonlexic

[ IDs in alt text ! ]

Digonlexic: a gender connected to or best described by the word digon.

Enneagonlexic: a gender connected to or best described by the word enneagon.

Gigagonlexic: a gender connected to or best described by the word gigagon.

Hectogonlexic: a gender connected to or best described by the word hectogon.

Etymology: digon / enneagon / gigagon / hectogon; “lexic” a suffix for lexegenders

@lexegender-archival , @radiomogai , @oneofmanyarchives

[ID: an orange line divider with a star covered in flame in the middle. End ID]

#Digonlexic #Enneagonlexic #Gigagonlexic #Hectogonlexic #lexegender #gender #neogender #xenogender #liom #mogai #🪼 creations #category: genders

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ohheyidothat · 2 years

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A Snub disphenoid, Cuboctahedron, Enneagonal Prism, and a Disdyakis Triacontahedron!

Learning shapes and their names was for kindergarten omg. I didn’t realize people went ahead and named every possible shape they could think of. More coming.

#ohheyisaidthat #funshapes #if the links every break: they just link to the wikipedia page for the shape! which is where the photos are from.

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twice-star · 4 years

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