141 lines
9.0 KiB
HTML
141 lines
9.0 KiB
HTML
<!DOCTYPE html>
|
||
<html lang='en'>
|
||
<head>
|
||
<title>gome — worst shape</title>
|
||
<meta charset='utf-8'/>
|
||
<meta name='theme-color' content='#efe5d7'>
|
||
<meta name='viewport' content='width=device-width, initial-scale=1.0, shrink-to-fit=no'/>
|
||
<link rel='icon' type='image/x-icon' href='../favicon.ico'>
|
||
<link rel='preconnect' href='https://fonts.googleapis.com'>
|
||
<link rel='preconnect' href='https://fonts.gstatic.com' crossorigin>
|
||
<link href='https://fonts.googleapis.com/css2?family=Lora:ital@0;1&display=swap' rel='stylesheet'>
|
||
<link rel='stylesheet' type='text/css' href='../css/style.css'/>
|
||
</head>
|
||
<body>
|
||
<header id='header'>
|
||
<nav>
|
||
<a href='..'>back to gomepage</a>—<a href='.'>journal</a>
|
||
</nav>
|
||
</header>
|
||
<main>
|
||
<article>
|
||
<h1 id='title'>Quest for the worst shape</h1>
|
||
<time datetime='Fri, 3 Feb 2023 23:30:00 CST'>3 Feb 2023, 11:30 PM</time>
|
||
<p>
|
||
In <a href='rhombic-dodecahedron.html'>Wednesday’s post</a>, I wrote that I enjoy learning about geometry on Wikipedia,
|
||
and about a very nice shape that I found there, the rhombic dodecahedron.
|
||
Another thing I enjoy doing from time to time is hunting for the most messed-up, the most wicked, the most cursed shapes possible.
|
||
So let’s explore some of the worst shapes I’ve found together.
|
||
</p>
|
||
<h3>Great grand stellated polydodecahedron</h3>
|
||
<figure>
|
||
<img src='https://upload.wikimedia.org/wikipedia/commons/thumb/3/34/Ortho_solid_016-uniform_polychoron_p33-t0.png/480px-Ortho_solid_016-uniform_polychoron_p33-t0.png' width='480' height='480' />
|
||
<figcaption>Image credit: <a href='https://www.software3d.com/Stella.php'>Robert Webb’s Stella</a></figcaption>
|
||
</figure>
|
||
<p>
|
||
We’re starting things off gently with a not-so-terrible shape.
|
||
This is a 4-dimensional polytope made of 120 great stellated dodecahedra (a self-intersecting dodecahedron with pentagrams for faces).
|
||
This one might give you a start if you bumped into it around a blind corner, but it’s more intimidating than truly depraved.
|
||
Once you get to know it, you might even regard it as beautiful in its own way.
|
||
</p>
|
||
<h3>5-orthoplex</h3>
|
||
<figure>
|
||
<img src='https://upload.wikimedia.org/wikipedia/commons/5/56/Pentacross_wire.png' width='480' height='480' />
|
||
<figcaption>Image credit: <a href='https://commons.wikimedia.org/wiki/File:Pentacross_wire.png#Licensing'>Claudio Rocchini</a></figcaption>
|
||
</figure>
|
||
<p>
|
||
The 5-orthoplex is not such a bad shape on its own.
|
||
It’s actually a fairly simple 5-dimensional shape, being the dual of the 5-cube.
|
||
It consists of 32 4-simplexes (the 4D extension of a triangle).
|
||
</p><p>
|
||
What makes this incarnation of it so truly awful is its tortured journey into our 3-dimensional world, not to mention the further bondage it endures to fit on your 2D screen.
|
||
To put it simply, this is a perspective projection of a stereographic projection of a Schlegel diagram of the 5-orthoplex.
|
||
I’ll spare you the gory details, but essentially, this shape has been put through a lot, and now it’s hungry for revenge.
|
||
</p>
|
||
<h3>600-cell, <span style='font-variant: small-caps;'>aka</span> hexacosichoron, hexacosihedroid, tetraplex, polytetrahedron, or C<sub>600</sub></h3>
|
||
<figure>
|
||
<img src='https://upload.wikimedia.org/wikipedia/commons/thumb/9/9a/600-cell_net.png/524px-600-cell_net.png' width='524' height='480' />
|
||
<figcaption>Image credit: <a href='https://www.software3d.com/Stella.php'>Robert Webb’s Stella</a></figcaption>
|
||
</figure>
|
||
<p>
|
||
We’re back down to four dimensions again, but this one brings a lot of cells to the table, as well as some freaky images.
|
||
This first one is the net of the 600-cell, which means it’s unwrapped to display the writhing mass of tetrahedra that make it up.
|
||
Here’s a projection of it rotating:
|
||
</p>
|
||
<figure>
|
||
<img src='https://upload.wikimedia.org/wikipedia/commons/1/1e/600-cell.gif' width='255' height='255' />
|
||
<figcaption>Look at it seethe.<br/>Image credit: <a href='https://commons.wikimedia.org/wiki/File:600-cell.gif#Licensing'>Jason Hise</a></figcaption>
|
||
</figure>
|
||
<p>
|
||
And to top it off, here it is forced through an unseemly chain of projections like the 5-orthoplex:
|
||
</p>
|
||
<figure>
|
||
<img src='https://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Stereographic_polytope_600cell.png/480px-Stereographic_polytope_600cell.png' width='255' height='255' class='no-border' />
|
||
<figcaption>There’s way too much going on in the center of this image.<br/>Image credit: <a href='https://commons.wikimedia.org/wiki/File:Stereographic_polytope_600cell.png#Licensing'>Althepal</a></figcaption>
|
||
</figure>
|
||
<h3>Crossed square antiprism</h3>
|
||
<figure>
|
||
<img src='https://upload.wikimedia.org/wikipedia/commons/5/52/Crossed_square_antiprism.png' width='360' height='403' class='no-border' />
|
||
<figcaption>Image credit: <a href='https://commons.wikimedia.org/wiki/File:Crossed_square_antiprism.png#Licensing'>Tomruen</a></figcaption>
|
||
</figure>
|
||
<p>
|
||
The simplest shape we’ve seen so far, but simply demented.
|
||
Why would you make a shape like that?
|
||
</p>
|
||
<h3>Boy’s Surface</h3>
|
||
<figure>
|
||
<img src='https://upload.wikimedia.org/wikipedia/commons/b/bb/Boy_Surface-animation-small.gif' width='400' height='400' />
|
||
<figcaption>Image credit: <a href='https://commons.wikimedia.org/wiki/File:Boy_Surface-animation-small.gif#Licensing'>A13ean</a></figcaption>
|
||
</figure>
|
||
<p>
|
||
This one gets points for the funny name alone.
|
||
This is the first non-polytope of the batch; it’s actually a result from topology.
|
||
Werner Boy was told by his teacher David Hilbert to prove that the projective plane could <b>not</b> be immersed in 3D space.
|
||
Instead, he ended up finding a way to do it, and this goofy shape is the result.
|
||
Strange and lumpy in all the wrong places, Boy’s Surface is more unpleasant and comedic than cursed.
|
||
</p>
|
||
<h3>Schwarz H minimal surface (Triply periodic minimal surface)</h3>
|
||
<figure>
|
||
<img src='https://upload.wikimedia.org/wikipedia/commons/thumb/7/79/Schwarz_H_Surface.png/500px-Schwarz_H_Surface.png' width='500' height='480' />
|
||
<figcaption>Evil baklava. Do not eat.<br/>Image credit: <a href='https://commons.wikimedia.org/wiki/File:Schwarz_H_Surface.png#Licensing'>Anders Sandberg</a></figcaption>
|
||
</figure>
|
||
<p>
|
||
Topology really takes the cake for making awful shapes no one ever wanted to look at.
|
||
I’m not even sure what goes where here.
|
||
With the normal geometry stuff, I could at least follow a general idea of what the shape was supposed to be.
|
||
Now I feel like the shapes are mocking me.
|
||
Wish I could provide more detail on the math here, but I’m out of my depth.
|
||
</p><p>
|
||
I could include many more horrible shapes here, but it’s getting late.
|
||
I’ve already spent way more time on this post than I should have.
|
||
So get ready, because here comes my pick for <i><b>the</b></i> worst shape ever:
|
||
</p>
|
||
<h2>Worst Shape Ever</h2>
|
||
<h3>Alexander horned sphere</h3>
|
||
<figure>
|
||
<img src='https://upload.wikimedia.org/wikipedia/commons/thumb/0/0a/Alexander_horned_sphere.png/640px-Alexander_horned_sphere.png' width='640' height='480' />
|
||
<figcaption>No one deserves credit for this abomination. (just kidding, the image is uncertainly attributed to a user named BernardH but it’s public domain)</figcaption>
|
||
</figure>
|
||
<p>
|
||
Just look at that. Simply criminal. Even by mathematicians, this shape is considered pathological.
|
||
</p><p>
|
||
The pattern you see here goes on forever.
|
||
To keep making this disaster of a shape, you have to keep removing sections and replacing it with a recursive pattern, forever.
|
||
An ouroborous, twisting back to bite its own tail but never reaching, gnashing its teeth and fading away.
|
||
</p><p>
|
||
Once again, we have topologists to blame for this one, namely James Waddell Alexander II.
|
||
I bet he’s proud of this nonsense.
|
||
The really galling thing about it is that topologically, this thing is equivalent to a sphere.
|
||
The friendly, puffy shape we all know and love, and this wicked, wicked shape claims relation to it.
|
||
Truly the height of hubris (or perhaps the supremum of hubris for you depraved topologists).
|
||
</p><p>
|
||
Do you know of any other awful shapes?
|
||
Have you seen a worse shape than the Alexander horned sphere? (not likely)
|
||
Do you know of any wholesome shapes that may help me recover from this dark path I’ve lost myself on?
|
||
Let me know your thoughts at my Ctrl-C email: <code>gome<span style='user-select: none;'> ​</span>@<span style='user-select: none;'> ​</span>ctrl-c.club</code>.
|
||
</p>
|
||
</article>
|
||
</main>
|
||
</body>
|
||
</html>
|