Ditopes, hosotopes, what's in between?

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

Ditopes, hosotopes, what's in between?

Postby mr_e_man » Mon Jan 31, 2022 4:44 am

Suppose P is an abstract N-dimensional polytope. To say that Q is a [j, k]-subpolytope (of P) means that the nulloid of Q is a j-face of P, and the body of Q is a k-face of P. In the style of Bowers acronyms, I might abbreviate this as a [j,k]-spyt (sub-polytope). For example, a [0, N]-spyt is a vertex figure, a [1, N]-spyt is an edge figure, a [k-1, k+1]-spyt is a k-dyad, a [-1, N]-spyt is the whole polytope P itself, and a [-1, k]-spyt is what is often called a k-face (though a k-face is actually only a single element, not a collection of elements as a polytope is supposed to be).

A hosotope is an N-polytope where every [-1, 2]-spyt is a digon. Equivalently, it has only two 0-faces.

A ditope is an N-polytope where every [N-3, N]-spyt is a digon. Equivalently, it has only two (N-1)-faces.

Generalizing, for any k, we can consider an N-polytope where every [k, k+3]-spyt is a digon. Equivalently, every [-1, k+3]-spyt is a ditope. Equivalently, every [k, N]-spyt is a hosotope.

So, what does a polychoron look like, where every [0,3]-subpolytope is a digon? It resembles a duoprism, as it has the same symmetry. There are n vertices, n edges, m faces (2-faces), and m cells. Each vertex figure is an m-gonal hosohedron, each edge figure is an m-gon, each face (rather [-1,2]-spyt) is an n-gon, and each cell (rather [-1,3]-spyt) is an n-gonal dihedron. In fact this is just (Schlafli) {n,2,m}.

Obviously, plenty of examples are provided by the regular polytopes with '2' anywhere in the Schlafli symbol.

For irregular examples, we can take any irregular polyhedron, make a dichoron, and then build a polyteron using such pieces (e.g. a diteron, but preferably something more interesting), so that every [1,4]-spyt is a digon.
ΓΔΘΛΞΠΣΦΨΩ αβγδεζηθϑικλμνξοπρϱσςτυϕφχψωϖ °±∓½⅓⅔¼¾×÷†‡• ⁰¹²³⁴⁵⁶⁷⁸⁹⁺⁻⁼⁽⁾₀₁₂₃₄₅₆₇₈₉₊₋₌₍₎
ℕℤℚℝℂ∂¬∀∃∅∆∇∈∉∋∌∏∑ ∗∘∙√∛∜∝∞∧∨∩∪∫≅≈≟≠≡≤≥⊂⊃⊆⊇ ⊕⊖⊗⊘⊙⌈⌉⌊⌋⌜⌝⌞⌟〈〉⟨⟩
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Re: Ditopes, hosotopes, what's in between?

Postby mr_e_man » Mon Jan 31, 2022 6:27 pm

mr_e_man wrote:Generalizing, for any k, we can consider an N-polytope where every [k, k+3]-spyt is a digon. Equivalently, every [-1, k+3]-spyt is a ditope. Equivalently, every [k, N]-spyt is a hosotope.

Here's another equivalent description: For any flag, the rank k+1 flag move commutes with the rank k+2 flag move. It follows that, for any j≤k+1 and any l≥k+2, the rank j flag move commutes with the rank l flag move.

Here's another equivalent description: The polytope "splits" as some kind of product A×B, where A is a [-1, k+2]-spyt and B is a [k+1, N]-spyt. viewtopic.php?p=28303#p28303
ΓΔΘΛΞΠΣΦΨΩ αβγδεζηθϑικλμνξοπρϱσςτυϕφχψωϖ °±∓½⅓⅔¼¾×÷†‡• ⁰¹²³⁴⁵⁶⁷⁸⁹⁺⁻⁼⁽⁾₀₁₂₃₄₅₆₇₈₉₊₋₌₍₎
ℕℤℚℝℂ∂¬∀∃∅∆∇∈∉∋∌∏∑ ∗∘∙√∛∜∝∞∧∨∩∪∫≅≈≟≠≡≤≥⊂⊃⊆⊇ ⊕⊖⊗⊘⊙⌈⌉⌊⌋⌜⌝⌞⌟〈〉⟨⟩
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