Hyper-Octagons

[Polyhedron Dude]

Most of us are familiar with the hyper-cubes (dyad, square, cube, tesseract, penteract, hexeract, etc.), and the hyper-triangles (dyad, triangle, tetrahedron, pentachoron (5-cell), hexateron, heptapeton, etc), so I thought I'll introduce the hyper-octagons. We'll start with the octagon.

To form an octagon from a square, we could use two methods - we could cut the corners off the square, or we could blow the edges of the square outwards and fill in the gaps with more edges - we'll use these methods on the hypercubes.

3-D: we start with a cube and cut the corners off - this leads to a figure with 6 octagons and 8 triangles as faces - it is called a truncated cube, it's one of the Archimedian solids - I call it "tic" for short. Tic is sort of a blocky looking polyhedron. Tic can also be formed by taking a cube and blowing its edges out and filling in the gaps. But there is another figure that can be formed, by blowing out the 6 square faces of a cube and filling in the gaps with 12 more squares and 8 triangles - this figure is called the (small) rhombicuboctahedron - I call it sirco. It is also an Archimedian solid. Sirco looks more like a 3-D octagon than tic does - tic looks kind of blockish, while sirco looks quite octagonish. Sirco can also be formed by taking a cube and shaving the edges off and chopping the corners off too.

4-D: Starting with a tesseract, we can chop the corners off - this leads to the truncated tesseract - I call it tat. It has 8 tics and 16 tets (tetrahedra) for cells - it is very blockish - It would probably be the dullest polychoron for a tetronian polychoron-modeler to make - the 8 tics in it make up nearly the entire polychoron - almost turning it into a tesseract, with only 16 small gaps to fill in with the small tets. Tat can also be formed by blowing out the 32 edges of the tesseract. We can also blow out the 24 square faces of the tesseract and get a figure with 8 sircoes, 16 octs (octahedra), and 32 trips (triangular prisms) for cells - the sircoes connect to each other by the squares and are quite large - this figure is called the cantellated tesseract - also known as the small rhombated tesseract - I call it srit - an abbreviation of the latter name. Srit would be far more appealing as a tetronian model than tat would - but it still has a blockish appearance - it can also be formed by shaving off the edges and corners of a tesseract. Finally we take the tesseract and blow out its 8 cube cells - filling in the gaps with 24 more cubes, 32 trips, and 16 tets - this figure is the 4-D version of the octagon - if a tetronian held it and looked strait at one of it's primary cubes (original eight) the "outface" (hyper-outline) would look like a sirco. This figure is called the small disprismatotesseractihexadecachoron (WHAT!) - it's also called a runcinated tesseract - I prefer calling it a sidpith (whew - sounds better). A sidpith would make an attractive and simple tetronian model.

I would challenge the readers to try to find the 5-D hyper-octagons - there are 4 of them. My short names for them are tan, sirn, span, and scant - can you figure out what they look like?

Polyhedron Dude

[Keiji]

how nice for you, i'll bet the first is one with its edges blown out, the second is one with its squares blown out, the third is with its cubes blown out, and the fourth is with its tesseracts blown out.

and the first would look mostly like a 5-d cube, whereas the last would look like a 5-d sphere.

well that took a lot of effort to think of

[Polyhedron Dude]

Yep, got them right - this sequence continues up the dimensions, adding one more each time - in 6-D there would be 5 of these, in 50-D there would be 49 - the first looking the most blockish, the last looking the most octagonish.

[BClaw]

I like your explanation Polyhedron Dude, I think I'm going to try and model a few of those and render them!