4th and 5th Dimensional Polyhedra

[GnomeToys]

 

http://www.polytope.net/hedrondude/polychora.htm

 

I can't tell if this is art or math or both, which means probably both.  This guy has been working on finding certain sets of 4D / 5D objects since the 90s with what appears to be the obsession of a total madman (aka great artist / mad scientist, whatever).  He has the following to say:

 

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My polychoron search began back in 1990, when I searched for them using vertex figures (verfs), faceting techniques, and a "digging-in-the-verf" technique. I used a blue note book and filled it with verf drawings, long names (many have changed since then), and cell lists. I carried that blue note book with me nearly every day to college, either to show people or to write more info into it. By 1993 there were over 1000 polychora in that book, although many of them were fissary or exotic-celled. That year also brought the discovery of the first (and presently only) non-prismattic uniform polychoron known to contain a snub polyhedron - Rapsady - rapsady contains 120 sirsids (also known as yog-sothoths), 120 sesides, and 1440 paps (pentagonal antiprisms). Later I started to hand draw sections of some of the more simpler polychora, mainly pentachorics and tessics, and wound up with approximately 100 polychora in hand drawn sections. Also during that time, I invented my short names (which have lately been called Bowers Style Acronyms by Richard Klitzing). These short names were the result of writing the polyhedra in an abbreviated form, where I later mentally pronounced the abbreviation, this lead me to change the long name abbreviation into a pronouncible short name. An example is the quasitruncated small stellated dodecahedron - abbreviated to QTSSD, mentally pronounced as quit SIS sid, later spelled as quit sissid. I now have short names for all of the 1849 known uniform as well as the additional scaliform polychora with the exception of many idcossids and dircospids. Also all uniform polyhedra and many of the uniform polytera (5 dimensional) have short names.

 

It wasn't until 1997, that I contacted other polyhedronist, starting with the legendary Magnus Wenninger, after that contact, I got a letter from another polyhedronist Vincent Matsko who got wind of my discoveries, this letter came immediately after I discovered the massive idcossid and dircospid regiments (as well as their lesser counter parts, the sidtaps and gidtaps) which is one of the biggest discoveries so far. Not long after that, I searched the web for 4-D polytopes and found George Olshevsky's web site and later contacted him. He also got wind of my discoveries before hand, so we compared info and teamed up to search for more polychora. In 1998 I discovered the blends (also known as the sabbadipady regiment), George later found Sto and Gotto, two members of the rit (rectified tesseract) regiment that have demitessic symmetry. George and I later allowed for coinciding cases which brought the polychoron count into the 8000s. In 1999 George found the most unusual polychora to date, the swirlprisms and later I started to investigate another unusual type of 4-D figure which I call polytwisters which are related to Hopf fibration, there are now 27 known regular polytwister plus an infinite group of regular dyadic twisters (also called "dysters"). Not long after this I created my first website. Also during this time I began my polychoron sectioning with POV-Ray, and eventually rendered sections of over 1000 polychora. John Cranmer has volunteered to make scores of polychoron section movies using my POV code, which he has placed on CDs, these were spectacular - however each file is several megabytes, so I can't fit them on my site.

Anyway aside from the objects themselves being very cool looking if you enjoy geometric patterning, what's equally impressive is that part where he developed what is basically a human-pronounceable language to describe the features of objects that don't exist as far as anyone can tell, and is apparently known in whatever esoteric field of mathematics this is for that along with all of these shapes.  The names of the objects are bizarre, but how exactly do you describe a dimension you can only view slices of?

 

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The name "dircospid" came from the first one I named, "dircospid pidsiddip piscax", short for diretrocuboctisnub prismatodiprismatodouble snub diakis prismatopseudosnub chiro600 which is an exotic celled one.

Anyway, I found the weird geometry pretty cool.  It would be fun to build one of these 3D slices as a sculpture (or multiple slices of the same 4D shape as a progression of sculptures) if I had that much motivation.  ? 

[Fist 666]

I've got to learn a whole lot of new vocab words to comprehend an nth of this.

 

I find conversations like this really fascinating, but just out of reach. My  Calc 2 professor at CU Boulder would sometimes spend half the class period on tangents about theoretical numbers and infinities and things of this sort, and while I was totally sucked into what he was talking about, I don't think I understood much of it. 

 

TLDR this is beyond me, but fascinates the fuck out of me.

 

 

[GnomeToys]

I studied mathematics but it doesn't help much with the higher dimensional stuff in terms of visualization...  the easiest way to interpret pictures is that the 4D ones are 3D "slices" along the 4th axis.  You can move on that axis, but that's it. 

 

With 5D you get an extra coordinate to mess with, so you're looking at something like a line through a 5D object instead, which can have either coordinate changed.  He displays that in the x/y style graphs of the 5D objects.  http://www.polytope.net/hedrondude/podiverts5.htm

 

In 6D it gets more confusing because you have to get rid of 3 axes, and with 7D and above you end up "collapsing" more dimensions than you started with to get any kind of image.  Wikipedia has this projection of a 9D cube to give an example of how silly things get.  https://en.wikipedia.org/wiki/Nine-dimensional_space#/media/File:9-cube_t0.svg

 

I'd have to read a bunch of math that wasn't in my field to understand how he might be finding these, and honestly my head is filled with enough abstract algebra that only sees real use in bizarre cryptography or reading more math papers to dive in that deep.  ?

 

 

 

[glorydays]

this is one of the more interesting sub-cultures ive only really glanced at

 

My glancing interest in this stuff came from reading HP Lovecraft novels. He places a lot of emphasis on mandalas and shapes that hold religious meanings.

[misteraven]

Currently feeling overwhelmed just reading this in two dimension reminds me of how complicated my three dimensional reality is, perhaps making me somewhat 1 dimensional as a person, so I'm not sure I have the fortitude to complicate my life by digging up additional dimensions.

 

Interesting conversation though.

[GnomeToys]

Yeah it's a very bizarre thing to think about.  One of my weirdest experiences involved "dreaming" in large dosages ? and I wound up being able to see in the 6th dimension.   Just picture an array of 4th dimensional time-space lines that all contain independent versions of the 3D universe we live in and the entire history of it, and a bunch of bored...  "entities"...  sitting around and messing with the parameters of them for fun.  It was sort of like Schroedinger's cats playing with an infinite number of strings.  They handed me one and told me I could mess with it, but every possible modification led to it turning into something worse than it already was so I stopped around the time I was trying to remake the world wars into something else.  After a while that retreated back into being able to see 4th dimensional strands and play with them in normal 3D space which was a lot less confusing.