It looks like there are two places on the wiki page that mention these conditions. The first is at the top of the page, and includes the self-dual condition: In fact, the 24-cell is the unique convex self-dual regular Euclidean polytope in [sic] which is neither a polygon nor a simplex. The second, which Fred quoted, is further down on the page and omits this condition: The 24-cell is the unique convex regular Euclidean polytope that is neither a polygon nor a simplex. Relaxing the condition of convexity admits two further figures: the great 120-cell and grand stellated 120-cell Apparently this page could use some work. Tom Adam P. Goucher writes:

Self-dual.

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Sent: Monday, March 09, 2015 at 1:36 AM From: "Fred Lunnon" <fred.lunnon@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: [math-fun] 24-cell --- it sez 'ere ...

At https://en.wikipedia.org/wiki/24-cell we are informed that

<< The 24-cell is the unique convex regular Euclidean polytope that is neither a polygon nor a simplex. Relaxing the condition of convexity admits two further figures: the great 120-cell and grand stellated 120-cell >>

The first sentence is plainly false as it stands (eg. octahedron, tesseract); the second (sic: missing period) yields me no clue as to what might instead have been intended by the first.

Any ideas out there?

WFL

Adam P. Goucher <apgoucher@gmx.com> wrote: |Self-dual. Good point — had never seen that particular distinction before. For that matter, the triangle is the only self-dual convex Euclidean polytope that is *both* a polygon and a simplex. --Dan

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Fred Lunnon <fred.lunnon@gmail.com> wrote:

At https://en.wikipedia.org/wiki/24-cell we are informed that self-dual << The 24-cell is the unique^convex regular Euclidean polytope that is neither a polygon nor a simplex. . . .

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The first sentence is plainly false as it stands (eg. octahedron, tesseract); . . . Any ideas out there?

Yes, indeed. --Dan

On Mar 8, 2015, at 10:11 PM, Tom Karzes <karzes@sonic.net> wrote:

Isn't a triangle the only proper polygon that is also a simplex, with or without the self-dual condition?

Tom

Dan Asimov writes:

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Adam P. Goucher <apgoucher@gmx.com> wrote:

|Self-dual.

Good point — had never seen that particular distinction before.

For that matter, the triangle is the only self-dual convex Euclidean polytope that is *both* a polygon and a simplex.

--Dan

a) The 24-cell — a.k.a. {3,4,3} — is unique among all polytopes, and b) the 4-space it lives in is also unique among all Euclidean spaces in that it has more than 1 inequivalent differentiable structure. (But it's not content to just have a finite number of them (like all the n-spheres S^n for n >= 7) or even a countable number; it has *continuum many* — i.e., 2^aleph_0 — of them. And, the 4-sphere S^4 is the only sphere among all S^n for which it is not known whether it can have more than one differentiable structure.) So it's natural to ask if a) and b) are somehow directly related. I have no idea. But it sure would be cool if they were. Its 24 vertices can be thought of as the binary tetrahedral group 2T as a subgroup of the unit quaternion group S^3. The space of cosets S^3 / 2T is a 3-manifold that is the configuration space of a regular tetrahedron inscribed in a unit 2-sphere S^2 (or equivalently, centered at th origin of R^3). This is defined as the space of all possible rotational positions of such a tetrahedron, such that if two positions look the same (i.e., the 4 vertices are the same) then they are the same in the configuration space. --Dan