Naming those polytopes

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

Naming those polytopes

Postby Adam Lore » Wed Jan 23, 2008 5:42 am

Hello! I am new.

I must say, this forum is pretty freaking great. Glad to now be a part of it.


I have been wondering, and many of you seem to be highly knowledgeable of such things:

Is there any alternative nomenclature for the polyhedrons and other polytopes? Names that aren't as long and strictly technical sounding?

It seems like at least a bunch of the uniform polyhedrons and regular polychora deserve non-technical, unique names. Like the cube. or pyramid. instead of stuff like icosatetrachoron or truncated hyperrhombic-great-stellated dodecadodeca-icosadecachoron.

I have been considering coming up with some, but I wanted to see what was already out there..
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Postby Adam Lore » Wed Jan 23, 2008 5:48 am

I forgot to add:

I know of some isolated cases, like Buckminster Fuller's "Vector Equilibrium" for the Cuboct, and, you know, like Tesseract for hypercubes. But nothing I've come across attempts to give unique, non-technical names for a whole bunch of shapes, like in a particular group or something.

Except for Polyhedron Dude! It is awesome what you do!! The only thing is though, is that, well, they just sound.. made up.
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Postby wendy » Wed Jan 23, 2008 7:12 am

Welcome to the forum.

A lot of names are technical because it's mainly mathematicians who study it. A lot of the mathematical terminology crosses the natural meanings of words, which makes the subject somewhat more confusing.

We do have, eg 'polychoron' as being largely established opposite polyhedron.

I did a fair bit on this line in the polygloss, which does deal with the idea from an etymological view as well, largely because i needed a termonology that makes sense in six dimensions.

http://www.geocities.com/os2fan2/gloss/index.html

Not many of the figures, even in 3d have names. For example, the dodecahedron can be applied to any of the polyhedra with 12 sides. In fact the "decagonal prism" is a more precise term, since it refers only to a certian figure.

In higher dimensions, the ideas greatly spread out and grow, so it is rather a matter of finding a regular terminology for all of this.

I use several number systems (10, 120), so the idea of using numbers for names (eg fifhundchoron), is a matter of also selecting the base. Base 10 is less suited for this than other systems.

The cuboctahedron itself, is a figure that has two different reflexes in four dimensions (one with 20 vertices, the other with 24). The notion of 'vector equilibrum', branches into three or so parts in higher dimensions. Much is to be made of the role that the CO plays in 3d, because it is not replicated completely in any higher dimension (except, prehaps 5d and 8d).

Consider also, that there are many strange things in 4d that we have no way of seeing in 3d, such as the bicircular tegum, or the pentagon-pentagram tegum (which has 25 regular tetrahedra as faces).
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Postby Adam Lore » Fri Jan 25, 2008 11:07 am

Thanks, Wendy. I have a whole lot to learn about higher dimensional polytopes. All of that stuff sounds strange and exciting!

I totally understand the use of technical names. I think it is the most practical and obvious way to go, for sure. I really like a lot of the names, I didn't mean to sound too negative..

But I think from a hobbyist point of view, or someone merely admiring the beauty of these things, it would be sweet to have more managable terms.

Also, I wanted to add, I didn't mean to badmouth Polyhedron Dude in anyway! I love all of those names. It is an amazing feat!
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Postby wendy » Sat Jan 26, 2008 8:36 am

DinoGeorge (George Olshevsky) and HedronDude (Jonathan Bowers) have done a lot of work with specifically 4d things. For example, Jonathan has enumerate all of the uniform star-polychora.

The particular terminology (GO+JB) is based on very long names, but one picks out representative letter clusters to form short names, like "NATO" or the like. Even regular processes, like "pentagon -> pentagon prism" produces irregular forms. In lots of cases, greater clarity comes from using the dynkin symbols.

Whatever one might say of their names, they have used several interesting ideas to find these, eg use of facetings etc.

The technical terminology of higher dimensions is based on lots of random sightings, rather than a consistant overview of the subject. So the manner of finding a name is to find something similar in 3d, and then pick one of its names.

For example, the use of 'cell' to refer to what i call 'face', comes from projections of four-dimensional surfaces as a series of bubbles (cells), and then transferring this to a meaning that has no connexion to cell.

The natural meaning of cell is a solid in a tiling, regardless of meaning. For example, the hexagons on war-gaming boards are called cells, as are spaces where "cellular automata" live in. Norman Johnson felt the need to "invent" cellule to adopt cell's original meaning.

Other things are to be seen. For example, one can talk of "polyhedra", where "hedron" refers to a 2d element, but "dihedral angle" refers to an N-1 dimensional thing. Likewise, face / facet / facing can mean 2d or n-1d. This means, eg in 6D, that /hedron/ and /face/ freely mean 2d or 5d things.

What i did in the PG is to separate these so that the same word has the same meaning, either relative to 0d (bridges) or Nd (walls). We see eg, that in 2d, "line" can be either a bridge (rail-line, straight line), or a wall (line in the sand, dead-line, front line). In 4d, ground is 3-dimensional, and the "wall" functions become 2-dimensional (eg front-line in 4d wars would be 2-dimensional).

I myself an an ameture too. But i can see six dimensions, and i can do etymologies. Conventional terminology is quite silly when applied to this space, so i had to redefine words to what they commonly mean.
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Postby Adam Lore » Mon Jan 28, 2008 7:14 am

Thanks, Wendy, very informative.

May I ask what you mean when you say that you can "see six dimensions" ?
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Postby wendy » Mon Jan 28, 2008 9:01 am

See six dimensions = see things that are six-dimensional. It's a little hard, but it can be done!
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Postby Adam Lore » Tue Jan 29, 2008 9:50 am

How do you do that?
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Postby zero » Wed Jan 30, 2008 5:32 am

Whatever she says + practice! :!:
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Postby wendy » Wed Jan 30, 2008 10:23 am

There are some things you do to visualise the higher dimensions. It is long, but we can start.

Sight is not what it seems. We see patterns, and we make meaning for them. The eye darts from place to place to make the image, and we build the picture from images and experience.

When you look to higher dimensions, there is nothing to see. So you must in your mind, paint the picture, and then walk in it. In order to paint the picture you must grasp what is the real essence of things.

A knife cuts. It does this by making a plane (N-1) in a sweep of action (1d), so the blade is N-2. So this is how you make it.

A road is a route of time (1d), so it is 1d in four dimensions.

A picture and a map is a plane (N-1d), so you can see everything in that space. A map is the ground, a picture has a top and bottom. Things don't fall in a map, but do in a picture.

From here, make in your mind, maps and pictures of things. Make simple objects. Explore the map of your neighbourhood. Make things like houses, roads, railways and rivers, by realising what these are. Go into a house and explore it.

Some numbers, like 7 and 10, are squares, and in higher bases, these become larger numbers, like cubes: 18 and 30. A typical 4d person can freely pick of eighteen objects like we pick 7, and might use a counting system of 30.

Try riding a wheel in "BC" style = stand on its axle. Steer it.

Try looking at the stars. A truly interesting thing happens.

Do calculations. You need some mathematics to paint the picture. Some profound insights will substitute for maths.
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Re: Naming those polytopes

Postby Adam Lore » Wed Feb 13, 2008 1:33 am

I will practice. I used to be able to visualize things in my mind very well, but lately I have almost completely lost the ability to.
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Re: Naming those polytopes

Postby kingmaz » Tue Dec 09, 2008 8:59 am

I'm a bit with Adam on this as I am new to the field also.

For me it's a bit like the Periodic Table. Names of elements such as iron, gold and lead have been known as such for thousands of years and thus have adopted usage. Whereas, if we had discovered the Periodic Table before individual elements, we might refer to the elements in a purely systematic way, such as Element 28 or 2,4 (by table position) and so on.

For more complex derivatives (truncated, rectified etc) of polytopes, it's obviously a good idea to name these in a systematic manner. But for the more base polytopes there's no reason why easier names shouldn't co-exist with systematic ones. I know that people like Bowers have adopted a highly abbreviated nomenclature for complex polytopes and it may well be that this ends up being the system adopted.

Multiple names exist already such as the 600-cell, tetraplex (which I like) etc for the hexacosichoron.

For me, names such such cubinder and cylindrone are helpful to give me an idea of what these figures are like just from their name. I would go further by shortening the cubic pyramid to the cubamid for example. It enables me to postulate that 5d figures like a glominder or tesserone might exist (these may be impossible, I don't know).

It depends ultimately on which way you approach it, but I'm afraid the onus will be on high geometers and mathematicians to correct us amateurs when our simplifications lead to misunderstandings. As Wendy says, an equilateral dodecahedron and decagonal prism are two quite different solids.
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Re: Naming those polytopes

Postby Keiji » Tue Dec 09, 2008 11:45 am

I would go further by shortening the cubic pyramid to the cubamid


That's not such a bad idea, actually :D

But what's a tesserone? Are you referring to a tapered tesserinder (which would be a 6D shape, and called a tesserindrone) or a tapered tesseract (which would be a 5D shape, and based on your "cubamid" idea, would be a tessamid).

Or we could call them all cones, as a cone is exactly the same as a pyramid, except that the convention is to use "cone" for curved bases and "pyramid" otherwise. That way your tesserone would be a tapered tesseract, and the tapered cube would be called a cubone :D
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Re: Naming those polytopes

Postby kingmaz » Tue Dec 09, 2008 1:40 pm

I think I mean a tessamid :\
In any case in claiming that name as my idea :D as it's first sensible one I've had in more than 3d!

I was thinking that perhaps a "cubamid" is a cube which has pyramidal extrusion to a point, displaced in the w-axis and therefore a tesseract might have a similar tapering extrusion in the 5d-axis.

I wasn't even aware that tapered tesserinder would be 6d, let alone called a tessidrone. I think I may be trying to run before I can crawl here.
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