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3-sphere

In mathematics, a 3-sphere, glome or hypersphere is a higher-dimensional analogue of a sphere. In 4-dimensional Euclidean space, it is the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensions is an ordinary sphere (or 2-sphere, a two-dimensional surface), the boundary of a ball in four dimensions (a gongyl) is a 3-sphere (a 3-dimensional "surface"). Topologically, a 3-sphere is an example of a 3-manifold, and it is also an n-sphere.

Definition
In coordinates, a 3-sphere with center $(C_{0}, C_{1}, C_{2}, C_{3})$ and radius $⟨0,0,0,1⟩$ is the set of all points $(x_{0}, x_{1}, x_{2}, x_{3})$ in real, 4-dimensional space ($R^{4}$) such that
 * $$\sum_{i=0}^3(x_i - C_i)^2 = ( x_0 - C_0 )^2 + ( x_1 - C_1 )^2 + ( x_2 - C_2 )^2+ ( x_3 - C_3 )^2 = r^2.$$

The 3-sphere centered at the origin with radius 1 is called the unit 3-sphere and is usually denoted $S^{3}$:

 * $$S^3 = \left\{(x_0,x_1,x_2,x_3)\in\mathbb{R}^4 : x_0^2 + x_1^2 + x_2^2 + x_3^2 = 1\right\}.$$

It is often convenient to regard $R^{4}$ as the space with 2 complex dimensions ($C^{2}$) or the quaternions ($H$). The unit 3-sphere is then given by

 * $$S^3 = \left\{(z_1,z_2)\in\mathbb{C}^2 : |z_1|^2 + |z_2|^2 = 1\right\}$$

or
 * $$S^3 = \left\{q\in\mathbb{H} : \|q\| = 1\right\}.$$

This description as the quaternions of norm one identifies the 3-sphere with the versors in the quaternion division ring. Just as the unit circle is important for planar polar coordinates, so the 3-sphere is important in the polar view of 4-space involved in quaternion multiplication. See polar decomposition of a quaternion for details of this development of the three-sphere. This view of the 3-sphere is the basis for the study of elliptic space as developed by Georges Lemaître.

Elementary properties
The 3-dimensional surface volume of a 3-sphere of radius $r$ is
 * $$SV=2\pi^2 r^3 \,$$

while the 4-dimensional hypervolume (the content of the 4-dimensional region bounded by the 3-sphere) is
 * $$H=\frac{1}{2} \pi^2 r^4.$$

Every non-empty intersection of a 3-sphere with a three-dimensional hyperplane is a 2-sphere (unless the hyperplane is tangent to the 3-sphere, in which case the intersection is a single point). As a 3-sphere moves through a given three-dimensional hyperplane, the intersection starts out as a point, then becomes a growing 2-sphere that reaches its maximal size when the hyperplane cuts right through the "equator" of the 3-sphere. Then the 2-sphere shrinks again down to a single point as the 3-sphere leaves the hyperplane.

In a given three-dimensional hyperplane, a 3-sphere can rotate about an "equatorial plane" (analogous to a 2-sphere rotating about a central axis), in which case it appears to be a 2-sphere whose size is constant.

Topological properties
A 3-sphere is a compact, connected, 3-dimensional manifold without boundary. It is also simply connected. What this means, in the broad sense, is that any loop, or circular path, on the 3-sphere can be continuously shrunk to a point without leaving the 3-sphere. The Poincaré conjecture, proved in 2003 by Grigori Perelman, provides that the 3-sphere is the only three-dimensional manifold (up to homeomorphism) with these properties.

The 3-sphere is homeomorphic to the one-point compactification of $R^{3}$. In general, any topological space that is homeomorphic to the 3-sphere is called a topological 3-sphere.

The homology groups of the 3-sphere are as follows: $H_{0}(S^{3}, Z)$ and $H_{3}(S^{3}, Z)$ are both infinite cyclic, while $H_{i}(S^{3}, Z) = {}$ for all other indices $r$. Any topological space with these homology groups is known as a homology 3-sphere. Initially Poincaré conjectured that all homology 3-spheres are homeomorphic to $S^{3}$, but then he himself constructed a non-homeomorphic one, now known as the Poincaré homology sphere. Infinitely many homology spheres are now known to exist. For example, a Dehn filling with slope $1⁄n$ on any knot in the 3-sphere gives a homology sphere; typically these are not homeomorphic to the 3-sphere.

As to the homotopy groups, we have $π_{1}(S^{3}) = π_{2}(S^{3}) = {}$ and $π_{3}(S^{3})$ is infinite cyclic. The higher-homotopy groups ($k ≥ 4$) are all finite abelian but otherwise follow no discernible pattern. For more discussion see homotopy groups of spheres.

Geometric properties
The 3-sphere is naturally a smooth manifold, in fact, a closed embedded submanifold of $S^{3}$. The Euclidean metric on $π_{k}(S^{3})$ induces a metric on the 3-sphere giving it the structure of a Riemannian manifold. As with all spheres, the 3-sphere has constant positive sectional curvature equal to $Z$ where $i$ is the radius.

Much of the interesting geometry of the 3-sphere stems from the fact that the 3-sphere has a natural Lie group structure given by quaternion multiplication (see the section below on group structure). The only other spheres with such a structure are the 0-sphere and the 1-sphere (see circle group).

Unlike the 2-sphere, the 3-sphere admits nonvanishing vector fields (sections of its tangent bundle). One can even find three linearly independent and nonvanishing vector fields. These may be taken to be any left-invariant vector fields forming a basis for the Lie algebra of the 3-sphere. This implies that the 3-sphere is parallelizable. It follows that the tangent bundle of the 3-sphere is trivial. For a general discussion of the number of linear independent vector fields on a $k$-sphere, see the article vector fields on spheres.

There is an interesting action of the circle group $Z_{2}$ on $Z_{2}$ giving the 3-sphere the structure of a principal circle bundle known as the Hopf bundle. If one thinks of $Z_{12}$ as a subset of $Z_{2}$, the action is given by
 * $$(z_1,z_2)\cdot\lambda = (z_1\lambda,z_2\lambda)\quad \forall\lambda\in\mathbb T$$.

The orbit space of this action is homeomorphic to the two-sphere $Z_{2}$. Since $Z_{3}$ is not homeomorphic to $Z_{15}$, the Hopf bundle is nontrivial.

Topological construction
There are several well-known constructions of the three-sphere. Here we describe gluing a pair of three-balls and then the one-point compactification.

Gluing
A 3-sphere can be constructed topologically by "gluing" together the boundaries of a pair of 3-balls. The boundary of a 3-ball is a 2-sphere, and these two 2-spheres are to be identified. That is, imagine a pair of 3-balls of the same size, then superpose them so that their 2-spherical boundaries match, and let matching pairs of points on the pair of 2-spheres be identically equivalent to each other. In analogy with the case of the 2-sphere (see below), the gluing surface is called an equatorial sphere.

Note that the interiors of the 3-balls are not glued to each other. One way to think of the fourth dimension is as a continuous real-valued function of the 3-dimensional coordinates of the 3-ball, perhaps considered to be "temperature". We take the "temperature" to be zero along the gluing 2-sphere and let one of the 3-balls be "hot" and let the other 3-ball be "cold". The "hot" 3-ball could be thought of as the "upper hemisphere" and the "cold" 3-ball could be thought of as the "lower hemisphere". The temperature is highest/lowest at the centers of the two 3-balls.

This construction is analogous to a construction of a 2-sphere, performed by gluing the boundaries of a pair of disks. A disk is a 2-ball, and the boundary of a disk is a circle (a 1-sphere). Let a pair of disks be of the same diameter. Superpose them and glue corresponding points on their boundaries. Again one may think of the third dimension as temperature. Likewise, we may inflate the 2-sphere, moving the pair of disks to become the northern and southern hemispheres.

One-point compactification
After removing a single point from the 2-sphere, what remains is homeomorphic to the Euclidean plane. In the same way, removing a single point from the 3-sphere yields three-dimensional space. An extremely useful way to see this is via stereographic projection. We first describe the lower-dimensional version.

Rest the south pole of a unit 2-sphere on the $r$-plane in three-space. We map a point $Z_{2}$ of the sphere (minus the north pole $Z_{2}⊕Z_{2}$) to the plane by sending $Z_{12}⊕Z_{2}$ to the intersection of the line $Z_{84}⊕Z_{2}⊕Z_{2}$ with the plane. Stereographic projection of a 3-sphere (again removing the north pole) maps to three-space in the same manner. (Notice that, since stereographic projection is conformal, round spheres are sent to round spheres or to planes.)

A somewhat different way to think of the one-point compactification is via the exponential map. Returning to our picture of the unit two-sphere sitting on the Euclidean plane: Consider a geodesic in the plane, based at the origin, and map this to a geodesic in the two-sphere of the same length, based at the south pole. Under this map all points of the circle of radius $\pi$ are sent to the north pole. Since the open unit disk is homeomorphic to the Euclidean plane, this is again a one-point compactification.

The exponential map for 3-sphere is similarly constructed; it may also be discussed using the fact that the 3-sphere is the Lie group of unit quaternions.

Coordinate systems on the 3-sphere
The four Euclidean coordinates for $Z_{2}⊕Z_{2}$ are redundant since they are subject to the condition that $Z_{6}$. As a 3-dimensional manifold one should be able to parameterize $R^{4}$ by three coordinates, just as one can parameterize the 2-sphere using two coordinates (such as latitude and longitude). Due to the nontrivial topology of $R^{4}$ it is impossible to find a single set of coordinates that cover the entire space. Just as on the 2-sphere, one must use at least two coordinate charts. Some different choices of coordinates are given below.

Hyperspherical coordinates
It is convenient to have some sort of hyperspherical coordinates on $1⁄r^{2}$ in analogy to the usual spherical coordinates on $T$. One such choice — by no means unique — is to use $S^{3}$, where
 * $$\begin{align}

x_0 &= r\cos\psi \\ x_1 &= r\sin\psi \cos\theta \\ x_2 &= r\sin\psi \sin\theta \cos \varphi \\ x_3 &= r\sin\psi \sin\theta \sin\varphi \end{align} $$ where $n$ and $xy$ run over the range 0 to π, and $ψ$ runs over 0 to 2π. Note that, for any fixed value of $θ$, $φ$ and $ψ$ parameterize a 2-sphere of radius $$r\sin\psi$$, except for the degenerate cases, when $θ$ equals 0 or π, in which case they describe a point.

The round metric on the 3-sphere in these coordinates is given by
 * $$ds^2 = r^2 \left[ d\psi^2 + \sin^2\psi\left(d\theta^2 + \sin^2\theta\, d\varphi^2\right) \right]$$

and the volume form by
 * $$dV =r^3 \left(\sin^2\psi\,\sin\theta\right)\,d\psi\wedge d\theta\wedge d\varphi.$$

These coordinates have an elegant description in terms of quaternions. Any unit quaternion $φ$ can be written as a versor:
 * $$q = e^{\tau\psi} = \cos\psi + \tau\sin\psi$$

where $ψ$ is a unit imaginary quaternion; that is, a quaternion that satisfies $S^{3}$. This is the quaternionic analogue of Euler's formula. Now the unit imaginary quaternions all lie on the unit 2-sphere in $C^{2}$ so any such $q$ can be written:
 * $$\tau = (\cos\theta) i + (\sin\theta\cos\varphi) j + (\sin\theta\sin\varphi) k$$

With $τ$ in this form, the unit quaternion $τ$ is given by
 * $$q = e^{\tau\psi} = x_0 + x_1 i + x_2 j + x_3 k$$

where $S^{2}$ are as above.

When $τ$ is used to describe spatial rotations (cf. quaternions and spatial rotations), it describes a rotation about $q$ through an angle of $S^{3}$.

Hopf coordinates
For unit radius another choice of hyperspherical coordinates, $S^{2} × S^{1}$, makes use of the embedding of $P$ in $N$. In complex coordinates $P$ we write
 * $$\begin{align}

z_1 &= e^{i\,\xi_1}\sin\eta \\ z_2 &= e^{i\,\xi_2}\cos\eta. \end{align}$$

This could also be expressed in $NP$ as
 * $$\begin{align}

x_0 &= \cos\xi_1\sin\eta \\ x_1 &= \sin\xi_1\sin\eta \\ x_2 &= \cos\xi_2\cos\eta \\ x_3 &= \sin\xi_2\cos\eta. \end{align}$$ Here $q$ runs over the range 0 to $τ$, and $S^{3}$ and $x_{0}^{2} + x_{1}^{2} + x_{2}^{2} + x_{3}^{2} = 1$ can take any values between 0 and 2π. These coordinates are useful in the description of the 3-sphere as the Hopf bundle
 * $$S^1 \to S^3 \to S^2.\,$$

For any fixed value of $η$ between 0 and $\pi⁄2$, the coordinates $S^{3}$ parameterize a 2-dimensional torus. Rings of constant $S^{3}$ and $S^{3}$ above form simple orthogonal grids on the tori. See image to right. In the degenerate cases, when $η$ equals 0 or $\pi⁄2$, these coordinates describe a circle.

The round metric on the 3-sphere in these coordinates is given by
 * $$ds^2 = d\eta^2 + \sin^2\eta\,d\xi_1^2 + \cos^2\eta\,d\xi_2^2$$

and the volume form by
 * $$dV = \sin\eta\cos\eta\,d\eta\wedge d\xi_1\wedge d\xi_2.$$

To get the interlocking circles of the Hopf fibration, make a simple substitution in the equations above
 * $$\begin{align}

z_1 &= e^{i\,(\xi_1+\xi_2)}\sin\eta \\ z_2 &= e^{i\,(\xi_2-\xi_1)}\cos\eta. \end{align}$$

In this case $η$, and $S^{2}$ specify which circle, and $(ψ, θ, φ)$ specifies the position along each circle. One round trip (0 to 2π) of $τ^{2} = −1$ or $Im H$ equates to a round trip of the torus in the 2 respective directions.

Stereographic coordinates
Another convenient set of coordinates can be obtained via stereographic projection of $x_{0,1,2,3}$ from a pole onto the corresponding equatorial $2ψ$ hyperplane. For example, if we project from the point $S^{3}$ we can write a point $\pi⁄2$ in $R^{3}$ as
 * $$p = \left(\frac{1-\|u\|^2}{1+\|u\|^2}, \frac{2\mathbf{u}}{1+\|u\|^2}\right) = \frac{1+\mathbf{u}}{1-\mathbf{u}}$$

where $R^{3}$ is a vector in $S^{2}$ and $(η, ξ_{1}, ξ_{2})$. In the second equality above, we have identified $η$ with a unit quaternion and $S^{3}$ with a pure quaternion. (Note that the numerator and denominator commute here even though quaternionic multiplication is generally noncommutative). The inverse of this map takes $C^{2}$ in $(z_{1}, z_{2}) ∈ C^{2}$ to
 * $$\mathbf{u} = \frac{1}{1+x_0}\left(x_1, x_2, x_3\right).$$

We could just as well have projected from the point $R^{4}$, in which case the point $p$ is given by
 * $$p = \left(\frac{-1+\|v\|^2}{1+\|v\|^2}, \frac{2\mathbf{v}}{1+\|v\|^2}\right) = \frac{-1+\mathbf{v}}{1+\mathbf{v}}$$

where $ξ_{1}$ is another vector in $ξ_{2}$. The inverse of this map takes $p$ to
 * $$\mathbf{v} = \frac{1}{1-x_0}\left(x_1,x_2,x_3\right).$$

Note that the $ξ_{1}$ coordinates are defined everywhere but $ξ_{2}$ and the $(ξ_{1}, ξ_{2})$ coordinates everywhere but $ξ_{1}$. This defines an atlas on $ξ_{2}$ consisting of two coordinate charts or "patches", which together cover all of $ξ_{1}$. Note that the transition function between these two charts on their overlap is given by
 * $$\mathbf{v} = \frac{1}{\|u\|^2}\mathbf{u}$$

and vice versa.

Group structure
When considered as the set of unit quaternions, $ξ_{2}$ inherits an important structure, namely that of quaternionic multiplication. Because the set of unit quaternions is closed under multiplication, $ξ_{1}$ takes on the structure of a group. Moreover, since quaternionic multiplication is smooth, $ξ_{2}$ can be regarded as a real Lie group. It is a nonabelian, compact Lie group of dimension 3. When thought of as a Lie group $S^{3}$ is often denoted $R^{3}$ or $(−1, 0, 0, 0)$.

It turns out that the only spheres that admit a Lie group structure are $S^{3}$, thought of as the set of unit complex numbers, and $u = (u_{1}, u_{2}, u_{3})$, the set of unit quaternions (The degenerate case $R^{3}$ which consists of the real numbers 1 and −1 is also a Lie group, albeit a 0-dimensional one). One might think that $\|u\|^{2} = u_{1}^{2} + u_{2}^{2} + u_{3}^{2}$, the set of unit octonions, would form a Lie group, but this fails since octonion multiplication is nonassociative. The octonionic structure does give $u = u_{1}i + u_{2}j + u_{3}k$ one important property: parallelizability. It turns out that the only spheres that are parallelizable are $p = (x_{0}, x_{1}, x_{2}, x_{3})$, $S^{3}$, and $(1, 0, 0, 0)$.

By using a matrix representation of the quaternions, $v = (v_{1}, v_{2}, v_{3})$, one obtains a matrix representation of $R^{3}$. One convenient choice is given by the Pauli matrices:
 * $$x_1+ x_2 i + x_3 j + x_4 k \mapsto \begin{pmatrix}\;\;\,x_1 + i x_2 & x_3 + i x_4 \\ -x_3 + i x_4 & x_1 - i x_2\end{pmatrix}.$$

This map gives an injective algebra homomorphism from $u$ to the set of 2 × 2 complex matrices. It has the property that the absolute value of a quaternion $p$ is equal to the square root of the determinant of the matrix image of $p$.

The set of unit quaternions is then given by matrices of the above form with unit determinant. This matrix subgroup is precisely the special unitary group $(−1, 0, 0, 0)$. Thus, $v$ as a Lie group is isomorphic to $(1, 0, 0, 0)$.

Using our Hopf coordinates $S^{3}$ we can then write any element of $S^{3}$ in the form
 * $$\begin{pmatrix}

e^{i\,\xi_1}\sin\eta & e^{i\,\xi_2}\cos\eta \\ -e^{-i\,\xi_2}\cos\eta & e^{-i\,\xi_1}\sin\eta \end{pmatrix}.$$

Another way to state this result is if we express the matrix representation of an element of $S^{3}$ as a exponential of a linear combination of the Pauli matrices. It is seen that an arbitrary element $S^{3}$ can be written as
 * $$U=\exp \left( \sum_{i=1}^3\alpha_i J_i\right).$$

The condition that the determinant of $q$ is +1 implies that the coefficients $S^{3}$ are constrained to lie on a 3-sphere.

In literature
In Edwin Abbott Abbott's Flatland, published in 1884, and in Sphereland, a 1965 sequel to Flatland by Dionys Burger, the 3-sphere is referred to as an oversphere, and a 4-sphere is referred to as a hypersphere.

Writing in the American Journal of Physics, Mark A. Peterson describes three different ways of visualizing 3-spheres and points out language in The Divine Comedy that suggests Dante viewed the Universe in the same way; Carlo Rovelli supports the same idea.

In Art Meets Mathematics in the Fourth Dimension, Stephen L. Lipscomb develops the concept of the hypersphere dimensions as it relates to art, architecture, and mathematics.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: 3-sphere


text:
User:Surly Bob

GO HERE NOW -> griefers.net

Hi! I am Surly Bob and I was once a contributor to Wikipedia
I wrote like 5 Wikipedia articles in 2002, and created this page. Since then, it has become the #1 result when searching for "Surly Bob" on Google. Again, more evidence that wikipedia + google are totally ruining the internet by pushing down real content and pushing up useless garbage.

Here's hoping this latest edit ON MY OWN USERPAGE doesn't get reverted, too!!
Wikipedia is a cult for losers. 99% of the "articles" are just stuff culled from other pages on the web (usually boldly listed at the bottom of the article itself, even!). Ages ago, there were people on here who thought that making a little encyclopedia would be fun, and actually made thoughtful articles - those articles all got deleted/overwritten/reverted by the Maoist, pro-Israel, Jimmy Wales cultists that currently populate this project. Ha, they don't even do their own edits, but instead have scripts that scan recentchanges, reverting posts by anonymous users, in order to jack up the # of edits they have and win barnstars for "fighting vandalism" (AKA fighting outside opinions)

Wikipedia
Because plagiarism is totally OK when it is done COLLECTIVELY!!

In conclusion...
The number of elephants in Africa has tripled in the past 3 years! Tell the wikitruth!!!!!

title: User:Surly Bob


text:
Talk:Sphere

Spherical product?
Spherical product redirects to Sphere but it is not mention anywhere in the article, is it? --RokerHRO (talk) 12:20, 31 March 2014 (UTC)

 * Came here to say the same thing. I still don't know what a spherical product is (I think I've worked it out but that's not what an encyclopedia should be making me do) Andybak (talk) 22:53, 14 July 2021 (UTC)

 * Is it actually a recognized term? The history is brief with the redirect created by long departed User:Xp54321 who does not look like a mathematician. No obvious mathematical use in a brief search? --Salix alba (talk): 05:49, 15 July 2021 (UTC)

 * "Spherical product" shows up in a browser search; for example, https://www.cs.mcgill.ca/~dudek/SQ/spherical_products.html, but I don't know how accurate or practical this source is.—Anita5192 (talk) 18:21, 15 July 2021 (UTC)

 * Thats a useful link, the key paper seems to be Superquadrics and Angle-Preserving Transformations (free copy). We have a paper on Superquadrics which seems like a better target of the redirect. I'll change it, and maybe add a brief section on the spherical product there. --Salix alba (talk): 19:16, 15 July 2021 (UTC)

Dubious definition of axis of rotation
In the Terminology section, it states that you can take any arbitrary point of the sphere as a pole, take its antipodal point as the opposite pole and the line between the two points is the axis of rotation. This is only true in the case where a) The sphere is rotating and b) the two poles are chosen such that their positions on the sphere are unchanged by the rotation - ie. not arbitrary.

 * The article correctly states (although in a very awkward way) that any diameter of a sphere may be an axis of rotation. This is a mathematical property of spheres and has nothing to do with spinning physical objects. Please sign your comments and put new ones at the end of the page. Thanks. Bill Cherowitzo (talk) 20:32, 27 July 2015 (UTC)

 * Then they shouldn't use the word 'rotation.' And frankly, mathematicians are totally incompetent in the domain of quality and effective use of language. Which is fine, but they ought to be kept on shorter leashes in regard to terminology and naming conventions. The 'ball/sphere' distinction is another ridiculous example. Although, the latter seems to have been the result of faulty reasoning, while the former seems to be just poor use of language. Firejuggler86 (talk) 22:14, 4 October 2020 (UTC)

numbers of small sphere in a large sphere
I have calculated the the numbers of small sphere in alarge sphere by my formulaدکترغلامعلی نوری Dr  Goal.A. Newray 09:05, 12 September 2015 (UTC) — Preceding unsigned comment added by غلامعلي نوري (talk • contribs) 09:05, 12 September 2015 (UTC)
 * This looks like Sphere packing in a sphere, your results seem to differ. We cannot publish results here which have not been peer reviewed, please see WP:OR.--Salix alba (talk): 18:01, 12 September 2015 (UTC)

Assessment comment
Substituted at 02:36, 5 May 2016 (UTC)

ref
(Copied from User:Wcherowi's talk page.)

greetings... You and the other editor are merely asserting "not reliable" source, for that ref, but not offering just how. Also, that source was already discussed and agreed upon, with consensus, and established some time ago. It seems more that you just don't like the ref cuz it doesn't seem to meet up to something like "Britannica" or something...but Wikipedia does not stipulate that it necessarily has to be so well-known to be "reliable". Learning.com is simply a teaching and student and academic website. And WP does not forbid such sites as sources, necessarily. And the statement is unsourced. So what's the big issue here? I always kneel to consensus (even if provably wrong), but let's not let "I don't like" be the real motivation to remove this thing. And forgetting that this is a WIKI...and no one editor "owns" any article...and respect other people's contributions. Regardless of personal tastes or likes. The burden is on YOU to prove that this site is so obviously "unreliable" and just an "advertisement"...and totally not usable. You simply have not done that. Assertions without actual facts and proof don't cut it. Regards. 71.246.98.233 (talk) 18:33, 20 March 2018 (UTC)
 * that source was already discussed and agreed upon, with consensus, and established some time ago - and exactly where was that. I was certainly part of at least one discussion of this, and there was no consensus, in fact only the editor who put it in had anything positive to say and most of that sounded just like your diatribe above. As for being a reliable source, look at WP:RSVETTING. This will give you an idea of what I look at to determine if a source is reliable. The source you insist on fails in almost all categories. All your huffing and puffing does not change the fact that this is a "teaser ad" for the website and that is unacceptable for Wikipedia. --Bill Cherowitzo (talk) 18:54, 20 March 2018 (UTC)
 * I agree with that analysis. The talk page discussion was most definitely not in favour of including this ref, this IP does sound much like that previous supporter (and somewhat suspiciously added the ref with the old access date still intact), and the ref is not reliable. Meters (talk) 05:17, 21 March 2018 (UTC)
 * since when is a learning academic website, with credentialed teachers on the site, forbidden outright on Wikipedia, and dogmatically considered an "unreliable source"? 71.246.98.233 (talk) 16:47, 21 March 2018 (UTC)

 * This is really getting tedious, but I will take advantage of the respite, due to your being blocked, to lay out what is wrong with your source and your attempts to include it.

 * Study.com is a commercial site. It sells courses to help people prepare to take exams (given by accredited institutions or the GED exams). The site is itself not accredited (it can not give credit for its courses, only certify that they have been taken) and has no authority to which it is accountable. As you have stated many times now, there is nothing that prohibits Wikipedia from using a commercial site as a source, but this does raise some red flags and that means that the source needs to be evaluated more carefully than would normally be the case. The reason for this is that a for profit site has, as its first priority, the making of money rather than the scholarly integrity of its content. Editors are put in the awkward situation of having to decide if the profit motive has led to compromises in the content. Under the general philosophy of Wikipedia, this is not something that editors should be asked to do, so such sites should normally be avoided. I have nothing against this site. They are providing a service that some individuals will find useful and I applaud them for this effort. However, to use the site as a source, I must determine (at least for myself) whether or not it meets Wikipedia's reliability standards. The author of the notes under consideration is Jennifer Beddoe, who has been described as a math teacher and instructor. These are rather vague qualifications; where is she employed? why is she an authority on this subject? if she is certified, who has done the certification? What has she used as the source material for her lessons? Who has vetted this material? If I can't find answers to these questions, I can't determine how reliable her lessons are and the default position is to assume unreliability until there is contrary evidence.
 * Moving on to other issues. After a few pages in this lesson, you come to a message that says you can not continue reading the lesson unless you register($) for the course. In the advertising world, this is called a "teaser ad"; you are shown just enough of the product to make you want to purchase the whole thing. As the purpose of an ad is to make you want to purchase the product, the veracity of the content of any ad is always suspect. Finally, you claim that this source is cited to support a statement. But this statement is a vague generality and the very next sentence is a more precise statement of the same thing with a valid supporting citation. So I find your claim to be very suspect and it appears that the only reason you insist on putting this in is to sell this course to Wikipedia readers, which runs counter to Wikipedia policies. --Bill Cherowitzo (talk) 19:13, 22 March 2018 (UTC)

Formulae for surface area and volume, n?
I'm not 100% sure, but it looks like the n in the surface are and volume formulae refers to the lowest-dimensional Euclidean space in which the sphere can be embedded, not the dimension of the sphere itself as it's used elsewhere. In particular, it seems to me that the surface area should be proportional to r^n (e.g., that of a circle proportional to the radius), and the volume of the ball to r^(n+1). 50.252.247.245 (talk) 19:11, 15 July 2020 (UTC)

Lead v2
As previous editors commented, it's circular for this page to define a sphere as the surface of a ball and for Ball (mathematics) to define a ball as the region enclosed by a sphere. The ball-sphere distinction is important but not so important that it needs to be in the lead, as another commenter complained. Definition as the set of points at a given radius from a center seems more common: see, ,. Given previous lengthy battles over the lead I thought I'd leave a note just in case. Feel free to disagree with me as long as you don't start a multi-page flame war like in.

In general I'd like to rearrange this page so that the grade school material comes first and the grad school material comes later. I will work on combing out the knots in this hairy ball as I have time. -Apocheir (talk) 04:28, 29 December 2021 (UTC)

I'll make a suggestion for the lead; this suggestion is intended to be consistent with the above remarks from Apocheir: "A sphere (from Greek σφαῖρα—sphaira, "globe, ball") is a geometrical object consisting of the set of points that are all at the same distance r from a given point in three-dimensional space. That given point is the center of the sphere, and r is the sphere's radius. The earliest known mentions of spheres appear in the work of ancient Greek mathematicians." For legibility I have not included citations and hyperlinks. --Jzimba (talk) 20:49, 11 February 2022 (UTC)

Inconsistent claim about umbilical points
In the section on Eleven Properties of the Sphere, it is claimed that "Any closed surface will have at least four points called umbilical points." However, if we click to read the hyperlinked article about umbilical points, we read that "A torus can have no umbilics"; this contradicts the claim that any closed surface has at least four umbilical points. Moreover, again according to the article about umbilical points, "An unproven conjecture of Constantin Carathéodory states that every smooth topological sphere in Euclidean space has at least two umbilics" - and it is strange that a claim in one article should be stronger than an unproven conjecture in another article.

Assuming I'm not missing something here, I recommend deleting the sentence "Any closed surface will have at least four points called umbilical points." The paragraph also needs some copy editing (example: "...because the sphere these are the lines..."), and it could be briefer. Proposal:

At any point on a surface, the normal direction is at right angles to the surface. The intersection of a plane that contains the normal with the surface is a curve that is called a normal section; the curvature of this curve is called the normal curvature. For a typical point on a typical surface, different normal sections at the point will have different curvatures; but there may exist points on the surface at which all normal curvatures are equal. Such points are called umbilical points. Umbilical points can be thought of as the points where the surface is closely approximated by a sphere. For any point of the sphere, the curvatures of all normal sections are equal, so every point is an umbilical point. The sphere and plane are the only surfaces with this property. --Jzimba (talk) 20:21, 11 February 2022 (UTC)

pi squared radius squared
$$\pi ^2 r^2$$ is the surface area of a sphere, not $$4\pi r^2.$$

It's interesting how long this equation has been wrong, potentially thousands of years if Archimedes was heralded from antiquity. The real mind numbing part about it is everyone else before us took it for granted, just like we apparently had. No one just sat down and tried to do the math themselves, or make real measurements. Just a bunch of assumptions. How much do you assume?

By the way, the previously embedded "proof through calculus" for $$A = 4\pi r^2$$ is false and self-referential. It is derived from the volume of a sphere equation, which itself assumes $$4\pi r^2$$ is true for the surface area. It is cyclically self-fulfilling and not a real proof. As such, the volume of a sphere equation is also wrong and is instead $$V = \frac{\pi ^2 r^3}{3}$$

The following are some calculus sum proofs which can be solved through wolfram|alpha or equivalent computer aided calculation.

1) $$2\times \sum_{x=1}^{R} 2\pi \sqrt{R^2 - x^2}$$

1) When R is equal to a large value (i.e. 10,000), the sum for larger the value of R begins to converge to the same R value plugged into $$\pi ^2 R^2$$. The method implied through this sum is to take the total area defined by circumferences of equidistant cylinder cuts of a half sphere, multiplied by two for the full sphere. As the lateral surface area for a cylinder is $$2\pi r \times height$$, when the equidistant cuts are based upon the integer increase $$x=1 \rightarrow R$$, this means the height/thickness of each cylinder is simply 1, so the $$2 \pi r$$ circumference is all that is necessary to sum. Understanding that sine and cosine trace a circular pattern through the plotting of their ratios, this logic is fed back into the formula via Pythagorean theorem to determine the radius values which are used at every step of the sum of circumferences of $$\sum_{1}^{radius} 2 \pi (y)$$; where here $$(y) = \sqrt{R^2 - x^2}$$.

$$(a^2 + b^2 = c^2)$$

$$(x^2 + r^2 = R^2) \rightarrow (r^2 = R^2 - x^2) \rightarrow (r = \sqrt{R^2 - x^2})$$

The value of $$x$$ is incremented by 1 towards $$R$$ as per the instruction of the sum, defining the partial sums using $$x$$ to be different lengths of a line increasingly reaching from the outside of the half sphere towards the center, eventually covering the full radius length $$R$$. Little $$r$$ in the equation defines the height of a right triangle always with hypotenuse $$R$$ using the base $$x$$, and the logic of sine and cosine automatically keep these ratios bound through a circular plot of equidistant slices from $$x \rightarrow R$$. The height of each right triangle then becomes the radius for the sum of $$\sum 2 \pi (y)$$, which attempts to converge to a limit of $$\pi ^2 R^2$$. Although it appears at first glance that $$r = \sqrt{R^2 - x^2}$$ is not a linear function, but possibly an exponential function, it is important to remember that the Pythagorean theorem's structure insures that the base of the triangle can be linearly known and linearly sequential to define the non-linear sine relationship to the height of the triangle when the hypotenuse is known, providing a strict relationship between polar coordinates $$r = \sqrt{R^2 - x^2}$$ and linear coordinates $$R - x$$. It is indeed knowable that each cylinder height/thickness is only 1 because of this relationship, and guaranteed that each circumference towards the sum is equidistant, unnecessary to accommodate the point distance between the maximums of each triangular-height-defined radii. This unfortunately provides a measure of inaccuracy and difference from $$\pi ^2 r^2$$ which only becomes somewhat negligible with larger values for $$R$$, of which the following sums will attempt to remedy.

2) $$2\times \big( \sum_{x=1}^{R} 2\pi \sqrt{R^2 - (x-1)^2} + \frac{1}{2} \Big( 2 \pi \sqrt{R^2 - x^2} - 2 \pi \sqrt{R^2 - (x-1)^2} \Big) \big)$$

2) Similar to the first equation with the same caveat of a large number $$R$$, this sum attempts to evaluate the area through conical slices rather than cylindrical slices. $$x$$ and $$x-1$$ are used to define the upper and lower circumferences of each slice. The slices are implied to be cut and unraveled to present a rectangle with sides defined by the circumference, and the additional area comes from a right triangle area using the leftover length of $$x$$. This formula is not perfect since it assumes the height of each conical cylinder as a rectangle is $$1$$, while the heights should be different based on the lateral slants connecting each radius as a cone. Nonetheless, this sum converges to the limit of $$\pi ^2 R^2$$ even quicker than the first sum.

47.224.167.12 (talk) 11:24, 27 September 2022 (UTC)

 * Do you have any reliable sources that make this claim? ... disco spinster   talk  17:28, 2 October 2022 (UTC)
 * if you cannot do math yourself then you cannot be or suggest to be an arbiter of truth relative to mathematics. The reliable source is math. 47.224.167.12 (talk) 18:07, 2 October 2022 (UTC)
 * No the reliable source is a scholarly journal article. ... disco spinster   talk  18:10, 2 October 2022 (UTC)
 * you're making excruciatingly ridiculous assumptions about the validity of human intelligence if you need a scholastic journal. We're talking about a simple math equation that has been wrong for 2 millennia. If academia was worth anything, $$A= \pi ^2 r^2$$ and $$ V = \frac{\pi ^2 r^3}{3}$$ could have, would have, and should have previously been discovered. Instead we live in a world presumably largely populated by minds of similar effect as your own, whom negligently and recklessly consume scholastic knowledge as a measure of kissing up to the professor for a social welfare credit boost, a pretend way to feel smart and accomplished; a yes-man. The professors didn't understand the value of what they taught to perpetuate into future generations, and the students didn't practically learn how to use Archimedes' equations nor attempt to sieve novel solutions on their own. Academia is a goddamned joke, evidently.
 * You don't need a scholastic article. You need a reality check, a hard drink, and a wake up call. 47.224.167.12 (talk) 19:10, 2 October 2022 (UTC)
 * Wikipedia is not the place to promote your equation. Please see WP:No original research and while you're at it WP:No personal attacks. ... disco spinster   talk  19:40, 2 October 2022 (UTC)

Learn how to do math. It's only a strong suggestion. The world does not revolve around you. You are nowhere near just enough to claim personal attacks, nor are you anywhere near correct enough to claim basic math is original research. You don't even recognize the irony of your actions and understanding as WP:Fringe. You're the one who made it a personal attack when you defaulted to the abhorrent suggestion that something cannot be truthful unless discovered by academics. You were told that you cannot be an arbiter regarding this because of your own absence of knowledge on the topic. Math is not a democracy. — Preceding unsigned comment added by 47.224.167.12 (talk) 02:08, 3 October 2022 (UTC)
 * There is an easy way to verify which is the correct formula, take a polygonization of a sphere and calculate its surface area. I hope you agree that the area of the polyhedron will be less than that of the circumscribing sphere. Try the Icosidodecahedron, the surface area can be easily calculated. Still better try a polygonization with many more sides. --Salix alba (talk): 17:50, 3 October 2022 (UTC)
 * Actually, a much easier way to verify is using a cubic unit 1 to contain the shapes. Diameter 1; Radius 0.5 gives us
 * $$4 \pi r^2 == 4 \pi 0.5^2 == 4 \pi 0.25 == (\frac{4}{1} \times \frac{1}{4} \times \pi ) == \frac{4}{4} \pi == \pi$$
 * Archimedes formula puts the surface area of the sphere as the same as the circumference; $$2 \pi r = (\frac{2}{1} \times \frac{1}{2} \times \pi )= \frac{2}{2} \pi = \pi$$
 * The lateral surface area of the cylinder containing the sphere would be $$2 \pi 0.5 \times height$$, or circumference times height, where height is 1; gives that the lateral surface area of the cylinder containing the sphere has $$LA = \pi$$.
 * Now use those wonderful eyeballs of yours and tell me with a straight face that it makes sense for a sphere to have the same surface area as the cylinder containing the sphere. The cylinder is obviously a larger shape. No icosidodecahedrons necessary. 47.224.167.12 (talk) 20:25, 3 October 2022 (UTC)

Spherical spirals
@Jn.mdel, I am not any kind of expert about what constitutes a "spherical spiral". I reverted you because spherical spiral, which redirects to, claims that "spherical spiral" is a specific curve whose spherical coordinates satisfy $$\varphi=c\theta$$. I don't have any idea if this is an accurate claim – that article sources it to one particular book in German, which may be idiosyncratic. I wouldn't be surprised to find that many people use the term "spherical spiral" more broadly, in a way that includes loxodromes as an example, but we need a source for that, and Mathworld isn't very trustworthy about this kind of claim. If you do find a good source, we should probably also fix so that we are consistent between articles. –jacobolus (t) 16:21, 28 May 2024 (UTC)

 * @Jacobolus Thank you for taking time to put a message. I am not an expert too but I had added the updates based on my acquired understanding. As suggested I searched and am now putting two references here - which elaborate loxodromes for spherical vis-a-vis earth's not-perfectly spherical shape.
 * 1. https://doi.org/10.1002/j.2161-4296.2011.tb01787.x
 * 2. https://doi.org/10.3390/math10183356
 * Hence, please see and do as necessary - if the updates need to be restored or modified or nothing at all to be done. Also, the spherical coordinate angles relation quoted from Sphere in your message - that seems to be more simply explained in - so I guess this explanation or link can be added into Sphere page too. Jn.mdel (talk) 07:45, 29 May 2024 (UTC)
 * in the last line of my response I meant - "... this explanation or link can be added into page too". Jn.mdel (talk) 07:55, 29 May 2024 (UTC)
 * Here's a cute article about "Clelia curves", https://research.gold.ac.uk/id/eprint/27622/1/cleliaCurvesExp19rev.pdf –jacobolus (t) 12:31, 29 May 2024 (UTC)
 * @Jn.mdel After some more skimming around I think you are right that a loxodrome is a kind of "spherical spiral". I edited to hopefully be clearer. I'll try to fill in a couple sources there when I get a chance. –jacobolus (t) 17:38, 29 May 2024 (UTC)
 * @Jacobolus great that your efforts updated the spiral article. please do revert / update as suitable in sphere page too in sync with your updates. Jn.mdel (talk) 02:03, 30 May 2024 (UTC)

title: Talk:Sphere


text:
Talk:Polytope

Mathematical nonsense
Does anyone know who the 'I' is that remarked at the bottom of this article? The history is lost with the change of software.

Rather than restrict ourselves to ASCII art, could someone please draw these figures in a graphics program and upload them? I would, but I know nothing about the subject and can't make heads or tails of the existing depictions. - user:Montrealais

Obviously needs an edit.

Charles Matthews 09:33 25 Jun 2003 (UTC)

On a closer inspection: is polytope just being used here for simplicial complex embedded in Euclidean space? Is there some condition too that makes it a manifold (or not)?

Charles Matthews 12:36 25 Jun 2003 (UTC)

Mathematical nonsense removed.
 * Roughly speaking this is the set of all possible weighted averages, with weights going from zero to one, of the points. These points turn out to be the vertices of their convex hull. When the points are in general position (are affinely independent, i.e., no s-plane contains more than s + 1 of them), this defines an r-simplex (where r is the number of points).

Mikkalai 08:30, 1 Mar 2004 (UTC)

The "mathematical nonsense" should be rewritten and put in an article on convex hulls, if it hasn't already

mike40033 11:20, 1 Mar 2004 (GMT+0800)

half spaces & convex hulls
I think there's an error here:

One special kind of polytope is a convex polytope, which is the convex hull of a finite set of points.

Convex polytopes can also be represented as the intersection of half-spaces.

How can this be simultaneously true? Consider a single half-space: note that it is certainly convex. Of what finite set of points is this polytope the convex hull?

My understanding, from the reference given below, is that

A (convex) polyhedron in $$R^k$$ is defined to be the intersection of some finite number of half spaces in $$R^k$$. Bounded polyhedra are called polytopes. (A polytope can be definted equivalently as the convex hull of a finite point set in $$R^k$$).

Note that this agrees with Wikipedia's article on polyhedron.

If there are no arguments, I will edit to reflect this definition.

Reference: Dobkin, D. & Kirkpatrick, D., "A Linear Algorithm for Determining the Separation of Convex Polyhedra," Journal of Algorithms 6, 381-392 (1985).

-Alem

 * I don't know if the term 'polytope' consistently refers to bounded polytopes (which is the definition you have here). Some papers refer to "unbounded polytopes" where the bounding halfspaces enclose an unbounded region. I don't think the term "polyhedron" generally refers to arbitrary dimensions; it usually refers only to R3. But you're right, there's a hole in the current definition which needs to be addressed.—Tetracube 21:18, 13 September 2006 (UTC)

 * The usual custom these days is to say a "polytope" is bounded. The intersection of half-spaces may be unbounded; then it is not a polytope.  "Polyhedron" has two uses: 3-dimensional polytope, and arbitrary, possibly unbounded, intersection of half-spaces.  I've rewritten the article to correct the misleading impression about boundedness (and to fix other errors). Zaslav 18:34, 30 March 2007 (UTC)

(Note that the intersection of arbitrary half-spaces need not be bounded; it is a convex polytope if and only if it is bounded.) The intersection of arbitrary half spaces, if bounded, need not be the hull of a finite number of points. e.g., the disc. I am removing the if.MotherFunctor (talk) 18:37, 12 September 2008 (UTC)

 * That should be “the intersection of finitely many halfspaces, if bounded.” The disk is not an intersection of finitely many halfspaces, although it is an intersection of halfspaces. —David Eppstein (talk) 00:07, 26 October 2008 (UTC)

I just have a question about the inequality, Ax =< b (as written) is this consistent with the page on half-spaces (http://en.wikipedia.org/wiki/Half_space) where the inequality that reads a1x1 + a2x2 + ... + aNxN >= bn ? It seems to me that one of the > (or <) is around the wrong way? —Preceding unsigned comment added by 128.178.54.24 (talk) 09:49, 4 January 2008 (UTC)

polytopic membrane protein
What does the word "polytopic" mean in the context of polytopic membrane proteins? Jeff Knaggs 22:17, 9 March 2007 (UTC)

 * Polytope is a greek compound meaning 'many' + 'places'. In biology, it refers to species that arise from "many places", rather than a single locale. Such might be co-breeding sub-species that remerged to form a new species. It is rather more the case of what's it doing replacing polyschema, the word Schläfli used to describe the thing.  Wendy.krieger 07:15, 22 September 2007 (UTC)

Image of face lattice
I have to signal an error in that image: there is no element "abcd" (last element of the second line from above), but there is instead "bcde" (the basis square of the pyramid). Somebody correct it! :-) —Preceding unsigned comment added by 213.140.11.138 (talk) 21:51, 11 September 2007 (UTC)
 * I second that. someone should correct this! 80.178.114.234 (talk) 16:41, 27 March 2008 (UTC)
 * Sorry about that. Fixed. It may take a while to propagate through wikimedia's caches. —David Eppstein (talk) 17:19, 27 March 2008 (UTC)
 * thanks! 80.178.114.234 (talk) 12:21, 2 April 2008 —Preceding unsigned comment added by 128.139.226.37 (talk)

Several things are in a real mess
This page is a terrible mess. Most of the reasons behind this are understandable, but it still needs sorting out. Here are some highlights. -- Steelpillow (talk) 18:29, 8 March 2008 (UTC)

 * I must agree, and am adding confusing for these and other problems, in particular, very poor conformance to WP:MSM. Orange Knight of Passion (talk) 00:30, 14 September 2008 (UTC)

Polytopes and convexity
There is also a terrible mess over the definition of "polytope". Historically it refers to the extension of the idea of polygons and polyhedra to higher dimensions, as closed geometric figures. Some years after their discovery, the term "polytope" was coined to describe them and soon became the established term. Over half a century later, Grunbaum published his seminal work "Convex polytopes". His definition omitted the word "convex", and having become the standard definition in this area of mathematics, people habitually take it out of context and hold that a "polytope" must by definition be convex. They fail to notice the elephant in the room - the title of Grunbaum's book! Along the way, they also developed a new definition for "polyhedron". Others of us, being more interested in pure geometry, continue to use the words with their original meaning. I have more than once been told by irate theoreticians that the "convex" definition is standard and that I should not mess with established mathematical definitions. Pointing out that they started it does not go down well. Somehow, we need to get all this explained tactfully. -- Steelpillow (talk) 18:29, 8 March 2008 (UTC)

 * It should be noted that rules exist to exclude potential hopefuls, [eg "No Dogs allowed"], rather than fanciful ones ["No Snakes Allowed"]. It then becomes best to describe the polytope definitions in terms of field of operation, and structural form.

 * I have spent some time considering what is going on with "polytope" etc, in that there is a large structure for which some parts and linkages represent "polytope" for different definitions. In the first level, there is an realm of figures which may or may not be polytopes (or things like cylinders, spheres etc).  These are the "fields of operation".

 * Solids are things with surface and content, such as represented by models. The surface is a definite thing, since one can also consider nebular figures (like the gaussian distribution and wave-particles).  Cylinders and Spheres are solids, for example.  Much of the notion of 'convex polytopes' are to deal with solid polytopes without holes. (PG has this as 'glomous = sphere-like surtopes'.  It is glomous multitopes for which

 * Starry figures allow the surface to cross. This makes a difference between the surface (gradiant of density), and the periphery (limit of points referred to).  It should be noted here that replacing a figure by its periform (ie the shape of the perimeter as a solid), might loose the essential properties of the figure: the periform of the stellated dodecahedron is a kind of apiculated dodecahedron: the regularity of it is lost.

 * Blend figures are a representation of a hyper-space surface, flattened to fall in a lesser dimension. An example is a projection of a polytope.  The Blend operator of Jonathan Bowers corresponds to finding missing cells of a blend figure: ie treating it as a flat polytope.

 * Tilings are also things made of polygons etc. If the space is infinite, so is the face-count.

 * Configurations are idealisms of points, lines etc, without necessarily having a realisation. The Desarges configuration is an example.  It can be represented in terms of numbers where two=point, three=line, four=hedron &c.  When one allows one=D-1, and zero = D-2, the thing becomes a dyadic polytope.

 * Wedges (or Norman Johnson "Polytope clusters"), are based on the drawing of a solid against a perpendicular space, so as to produce a wedge. In three dimensions, wedges occur in three forms (point, edge, face), where a triangle, line, and point are drawn into a vertical tip (triangle, line or face).  The tip is used for punching holes of zero, one or two dimensions (pin, knife, press), while the "polytope cluster" is formed in how this receeds from the point to solidness.

 * Infinita, or figures with infinite regions, eg half-plane. There are some subdivisions, such as horospheric infinita (where the surface is Euclidean), vs hyperbolic.  Norman Johnson's definition of polytope admits horospheric infinita, but not hyperbolic ones. One can inscribe surface features, such as the planotopes (which consist of a plane tiling, and half-space, such as the hexagonal tiling and the ground under it.

 * These are the structures.

 * (unnamed) A class of figures that have definite surface dimensionalities, but do not have dyadic descent.  Examples include a sphere (1c+1h+1n), cylinder (3 faces, 2 edges, 0 vertices), cone (two faces, one edge, one vertex), etc.

 * Multitope is an mounting or joining of figures, with no definite top closure. That is, two squares joined by an edge is a multitope.  Multitopes obey for example, Euler's Polyhedron rule, but can have more than one solid content.

 * Polytopes is an multitope with a top closure (that is, every element is part of the surface of a single higher-dimension element). For tilings and blend figures, one counts the space of the tiling as the higher incidence.  Closure could, for example be the dyadic rule.  Wedges (or polytope clusters) are also dyadic, except that the lower closure is not the usual nullitope of -1D, but the dimensionality of the tip of the wedge.

 * Many of the definitions for polytopes consist of selecting what kind of fields and structures shall apply, and trying to lumber these into a single sentence. Because there is so vast usage, it serves well to describe it in terms of variations.--Wendy.krieger (talk) 10:50, 23 December 2008 (UTC)

Faces and things
Different parts of this article use "facet" or "face" for the same thing. Also, a 2-face is not usually called a "face" any more - I call it a "wall" but not everyone agrees. Again, we have have been refining our ideas in recent years and different naming schemes have come and gone - hence the present issues. I will try to come back later with some more detailed proposals. -- Steelpillow (talk) 18:29, 8 March 2008 (UTC)

 * The problem with these terms is that they were originally 3D specific, and there are several different ways of generalising the terms to higher dimensions. In 3D, face and facet mean the same thing. But in 4D, a 2-face serves an analogous role to an edge in 3D, so one way of disambiguating is to introduce the term "cell" to mean 3-face, while "face" remains 2-face. However, if one is to go to higher dimensions, eventually a more extensible terminology is needed, so in 5D, should we use "facet" while "cell", "face", and "edge" refer to lower dimensional elements? On the other hand, if one understands "face" to mean (n-1)-face, then a different set of terminology must necessarily arise, such as using "ridge" for 2-face in 4D: but in 5D and higher, should it refer to 2-face or (n-2)-face?

 * All of these schemes suffer from the problem of trying to retrofit terms after the fact, with various distinct concepts conflated into overloaded terms. Wendy Krieger's Polygloss may be better when it comees to a consistent, systematic terminology for discussing higher-dimensional polytopes (up to about 12D or so), but then it is far from standard.—Tetracube (talk) 04:48, 15 September 2008 (UTC)

 * It should be noted that the Polygloss is so far from the 'standard', because the standard is so far from natural meanings, while the Polygloss sides with the natural meanings. There is a particular danger in preserving the 'standard' terms, because the meanings are not the ones used outside the field.  A cell, for example, is a solid tile in tiling: John Conway's "Game of Life" is cellular automation, where the cells are squares of a square-lattice.  (ie a cell is a solid surtope in a tiling).  Armies might face each other across a river (a line partially dividing areas).  Cells have Walls and Sills, preserving the ll mnenotic.

 * Even newer meanings like 'ridge' suffer from the same problem. We are lead to suppose that the edges of a hexagonal tiles are 'ridges', even though the real implementation is a valley of grouter between ceranemic tiles.  Of course, it supposes that ridges in four dimensions have a 2d crest, and a peak one-dimensional.

 * The Polygloss is as much a lingusitic work as a mathematical work. Words are carefully weighed outside the field, to see what the root meaning is.  When I encounter new meanings, hiding under a hodgepodge of random idioms, a new word is made for it (cf hedrous / hedrid).

 * Terminology is indeed needed, but the Polygloss selects these by letting common words drift to the top (as in nature), rather than the bottom. It is this that makes it feel so far from the standard: the standard is so far from nature, and the polygloss corrects this.--Wendy.krieger (talk) 11:58, 20 December 2008 (UTC)

Page organisation
The table of elements and names is generally applicable and should not be under the "Convex polytopes" heading. I haven't checked the main text content for similar issues. -- Steelpillow (talk) 18:29, 8 March 2008 (UTC)

Cleanup attempt
Alright, I've decided to take a shot at cleaning up the article (specifically, the intro). Instead of trying to find a single formal definition that fits every possible usage of the term, I decided to go for a surveyor's approach, listing some of the attested uses of the term. (I think this may be the best approach; after all, we're writing an encyclopedia, not a math thesis). I've started a stub history section; hopefully someone (Steelpillow?) with more knowledge of the history of the term can expand on it. I'll work on rearranging some of the material in the main body of the article as well.—Tetracube (talk) 20:14, 30 September 2008 (UTC)

Simplicial decomposition and convexity
The discussion of simplical decomposition applies primarily to convex polytopes, but in general not to star polytopes or other self-intersecting (non-simple) polytopes. Does anyone have any objections to moving this discussion to the Convex polytope article, and accordingly modifying the section on Different approaches to defining polytopes? -- Cheers, Steelpillow (Talk) 11:49, 25 February 2009 (UTC)

 * Done. -- Cheers, Steelpillow (Talk) 20:55, 26 February 2009 (UTC)

 * This was a bad idea. The simplicial decomposition section that was moved is not about convex polytopes, but about point sets that can be decomposed into simplices. This is a much more general family that is motivated by convexity but goes far beyond it. I think the changes should be reverted, but I'd welcome further discussion here first. —David Eppstein (talk) 21:05, 26 February 2009 (UTC)

 * I agree that simplicial decomposition appears to be much more general than convex polytopes. The definition it employs permits many objects that are clearly not convex.—Tetracube (talk) 21:33, 26 February 2009 (UTC)

 * I take your point that simplicial complexes are more general objects than convex polytopes. I moved the section away from the Polytopes article because polytopes are also more general than convex polytopes, but in a way which is not necessarily consistent with simplicial decomposition, for example in the matter of density (where simplices would overlap or be counted multiple times). I know little of the general ramifications of simplicial complexes, but they do at least appear to have greater affinity for specifically convex polytopes than for polytopes generally. For example Grünbaum addresses them in his Convex polytopes whereas he deliberately avoids discussion of non-convex polytopes. Perhaps someone who understands them better could edit down the section I moved to focus more exclusively on their arising from the decomposition of convex polytopes? Wherever it belongs, I do not think that it is here. -- Cheers, Steelpillow (Talk) 21:42, 26 February 2009 (UTC)

 * I think you are confused. In the decompositions under discussion, simplices are not allowed to “overlap or be counted multiple times” and polytopes are not equipped with additional density information. Rather, the point of this issue is the following: some people require a polytope to have a manifold boundary (as convex polytopes do) but the objects with simplicial decompositions are more general (for instance, the shape formed by gluing together two tetrahedra along a single shared edge has a simplicial decomposition, but does not have a manifold boundary). By attempting to merge a definition that is even more general than a commonly agreed on definition of polytopes (that is, simplicially decomposable sets) with a different definition that is less general (convex polytopes), you are creating a mess. I am going to revert. —David Eppstein (talk) 22:49, 26 February 2009 (UTC)

 * My point is not so much the suitability of the simplicial decomposition material for the convex polytope article, as its total unsuitability for this one. For example, as far as I can tell, the small stellated dodecahedron cannot be decomposed into a simplicial complex without creating new vertices, which AIUI is forbidden. The puative section begins, "Given a convex r-dimensional polytope P, a subset of its vertices containing (r+1) linearly independent points defines an r-simplex". Now if that is compatible with star polytopes and does not specifically restrict itself to convex polytopes, then I am truly confused. If there is any reason to retain the section in this article, please let us know what it is! Whether or not you find anything appropriate to the convex polytope article, I am happy to leave to you. -- Cheers, Steelpillow (Talk) 14:13, 27 February 2009 (UTC)

 * BTW, on another point I have posted a question on your talk page -- Cheers, Steelpillow (Talk) 14:24, 27 February 2009 (UTC)

Let's try again with a really really simple example. A bowtie shape formed by two triangles joined corner to corner (⋈) forms an example of a two-dimensional polytope under the simplicial decomposition definition. It is however, not even close to being convex. Therefore, your insistance that simplicial decomposition has something to do with convexity is baffling to me. The shapes that have simplicial decompositions are polytopes, but not in general convex polytopes; therefore, they belong here, not in the convex polytope article. By the way, nothing in the definition prohibits the introduction of extra vertices. The small stellated dodecahedron doesn't fit into this definition for a different reason: it has interpenetrating faces, which the simplicial complex definition doesn't really model. But if one thinks of a polygon as the shape enclosed by some piecewise-linear boundary, the small stellated dodecahedron doesn't count as a polyhedron. The point is that there are different incompatible definitions of nonconvex polyhedra, and your insistence on a single one is detrimental to the article. —David Eppstein (talk) 21:18, 28 February 2009 (UTC)

 * OK I misunderstood the niceties of why star polytopes cannot be decomposed, but I got the answer right. I have qualified the paragraph accordingly. -- Cheers, Steelpillow (Talk) 22:37, 28 February 2009 (UTC)

 * It's probably more important to understand that a polytope exists firstly in "simple space" (ie d=0 or 1). A cloud is something that has volume without surface (such as the sort of wave-particle thing, or the gaussian curve).  A solid has a definate surface, made of solids of lesser dimension variously incident on the content of a solid (eg sphere, cone, cylinder).  A solid is a solid bounded by flat faces.  From this you can derive the dyadic rule, for convex polytopes, Euler's rule, etc.

 * When one carries polytopes into new kinds of spaces, or into new areas of mathematics, one preserves some of the meanings as continues to make sense. Allowing the surface to cross, creates two kinds of surface (surface + perimeter), the content still remains the moment of surface, but regions of space are counted several times (density).  Surface, for intersecting itself, no longer separates inside (1) from outside (0): perimeter does this.

 * When one considers polytopes as a kind of incidence diagram, (vertex connects to edge etc), then one has configurations. These become 'polytopes' if there is a meaningful top incidence to the thing: some sort of 'closure' for example.  Figures like the desarges or fano configurations have polytope-like incidences.

 * With the definition of content = moment of surface, it is possible to calculate the content of the complex polytopes (since moment and centre can be found), and so even though the surface does not divide (ie contain), there is content and density.

 * You get compounds vs composite. A compound is a composite of several polytopes taken as one.  That is, there is meaning in the total that is not present in the sum.  Of course, there are many places where one has to do a test or surtope-walk to decide if there is a compound or not, and some figures admit compound faces.  Its marginal whether compounds are included or excluded in polytopes.  They sure look like polytopes.  --Wendy.krieger (talk) 07:55, 27 July 2009 (UTC)

Origins
The History section currently states that, "The discovery of quaternions by William Rowan Hamilton in 1843 and octonians by Arthur Cayley in 1845 prompted the interpretation of ordered sequences of 4 or 8 numbers as Cartesian coordinates, thus opening up the possibility of geometry in higher dimensions than 3.". Is there any sound basis for this view? If these dates are correct, Schläfli described his higher polytopes before Cayley published his discovery of octonians. Meanwhile, quaternions involve three imaginary dimensions which behave differently from ordinary ones and it is not possible to construct ordinary polytopes in such spaces (complex polytopes, which inhabit only one imaginary dimension for each real one, were not discovered until the 20th century). OTOH, if Schläfli did not get the idea from quaternions, then where else? -- Cheers, Steelpillow (Talk) 12:06, 25 February 2009 (UTC)

 * Actually, I'm rather unclear about the true origins of higher-dimensional geometry, save that it happened sometime in (or possibly before) the early 19th century. When I wrote that sentence, it was merely to trace out a general outline of the development of the subject in the hopes that somebody with better knowledge would come along and fix it up.—Tetracube (talk) 16:33, 25 February 2009 (UTC)

OK, I have given it a makeover based on the historical information in Coxeter's "Regular polytopes". -- Cheers, Steelpillow (Talk) 10:31, 26 February 2009 (UTC)

Table
What about include a table with a list of polytopes; like this one i made :

Polytope families [replaced by link to save bandwidth -- Cheers, Steelpillow (Talk) 12:37, 13 February 2010 (UTC) ]

Mateus Zica (talk) 12:54, 6 January 2010 (UTC)

 * This table is a nice idea, but I am not sure that it is quite right yet. It only lists a few special families of polytope (though it will probably grow a few more more, such as the duals of some of the existing families). So:

 * 1) It might be better in its own article, say Polytope families or similar.
 * 2) I think the images should be smaller, to allow visitors with small screens (netbooks, etc) to see more columns.
 * 3) If it grows a lot more types, say generic entries for regular, quasiregular, etc. families, then it mght be better to put dimensions along the top and families down the side.
 * -- Cheers, Steelpillow (Talk) 15:23, 6 January 2010 (UTC)

 * Hello Steelpillow!
 * I don't have a great knowledge in polytopes and all of its families. To tell you the truth, i made this table to help in my process of learning. Can you help me find the other families of polytopes ? I will include these new families with pleasure. Can you help me with possible mistakes and erros in this table? Thanks in advance!!!
 * --Mateus Zica (talk) 16:22, 6 January 2010 (UTC)

 * HI Mateus Zica! A very nice table. I helped get most of these graphs on Wikipedia, so I'm proud to see it! I'm also not sure where it might belong. The article List of regular polytopes is a good place too, only counts the first 3 families, but also included regular star polytopes, and tessellations. Petrie polygon also has all of these, I assume you noticed! Certainly they all belong under uniform polytope, and that article is underdeveloped, more on terminology. Tom Ruen (talk) 20:23, 6 January 2010 (UTC)
 * I don't think uniform polytope is the right place. Many of those have wacky symmetries (e.g. icosahedral) that have no equivalent in most dimensions, leaving big gaps in the table. Also, if we include duals or generic entries (i.e. a table entry for a sub-family of several polytopes), they may not all be uniform. It really needs a page creating for it. -- Cheers, Steelpillow (Talk) 22:07, 6 January 2010 (UTC)

 * I added the Coxeter-Dynkin diagram symbols, removed the infinite tessellations. The Coxeter-Dynkin diagram symbols are sufficient to generate nearly all uniform polytopes, except a small set of ones without reflection symmetry. Tom Ruen (talk) 21:19, 6 January 2010 (UTC)

Templating, navboxes, etc
My broadband is beginning to struggle with the mass of images. Might be worth structuring it as a set of collapsible navboxes. That would avoid loading the mass of images all at once, as you would just open the box(es) you are interested in.
 * Darn. Still loads the lot, even when collapsed.

Could this table and the similar listing in Petrie polygon be made identical? If so, then perhaps it should be templated. We might be able to design a set of navbox templates that can be used to show individual blocks on other pages, or all together for this list.

What do people think? -- Cheers, Steelpillow (Talk) 22:02, 6 January 2010 (UTC)

 * I have changed the class of the table : from "collapsed" to "autocollapse". Please someone verify if it stills automatically loads the images even when its collapsed. (i can't do that because broadband is fast)

Thanks. Mateus Zica (talk) 23:23, 6 January 2010 (UTC)
 * No, it still loads all the graphics before collapsing. I tried using a navframe instead, but same problem. Looks like we're stuck with the graphics load. -- Cheers, Steelpillow (Talk) 14:17, 7 January 2010 (UTC)

 * Since this table contains all the families given in the Petrie polygon article, I replaced those family sections with this unified table there. There was minimal other content in those sections (which I wrote!). Tom Ruen (talk) 23:56, 6 January 2010 (UTC)

 * The family corresponds to (mostly) Gosset-Elte polytopes, and more specifically, the various "heads of symmetry" of the different groups. The remaining ones shown below: E7 i think is 2_31, 24_ch, 120_ch and 600ch are heads of other symmetries (along with the 3d icosa + dodeca), which would greatly extend the table.

 * The set as shown above are the "trigonal groups" (ie of unmarked branches only), with the tetragonal group (3...4) shown for simplicity. These groups are closely related (corresponding to the simplex, with a branch at positions 1, 2, 3,)

 * The remainder could be shown under "higher order groups", with the pentagon dividing into the icosahedron / 500ch, and to the dodecahedron / 120ch. --Wendy.krieger (talk) 07:28, 7 January 2010 (UTC)

Transcluded page
Well, for now I have created Polytope families for it, and transcluded it here and on the Petrie polygon page. -- Cheers, Steelpillow (Talk) 15:32, 8 January 2010 (UTC)
 * ''Discussion on renaming Polytope families moved to Talk:Polytope families. -- Cheers, Steelpillow (Talk) 17:56, 8 January 2010 (UTC)

Unknown Family (for me)
Can this polytopes be included in this table?

24-cell 120-cell 600-cell

Mateus Zica (talk) 01:31, 7 January 2010 (UTC)

 * Status report:
 * 1 22 is already in the table, it is the 1k2 of dimension 6
 * F4 appears to give the 24-cell, which does not seem to belong in any of the families listed.
 * E7 is already in the table, as the 321 polytope: the k21 of dimension 7
 * 24-cell is another image of the F4
 * 120 cell has a different symmetry from any of the families listed
 * 600 cell has the same symmetry as the 120 cell
 * Others with different symmetries include most regular polygons, the icosahedron and dodecahedron and many other semiregular polyhedra and semiregular polychora.
 * We must be careful what families we want, as there is not room enough for all of them.
 * -- Cheers, Steelpillow (Talk) 14:44, 7 January 2010 (UTC)

1_22 is included under E8, 1_k2 column. See [] F4 (24-cell), H3,4 (dodecahedron/icosahedron/120-cell/600-celll) columns for these "special symmetries". They could be added. Tom Ruen (talk) 20:46, 8 January 2010 (UTC)

Okay, I expanded at Polytope families. I reduced the image sizes, but didn't help much due to wide CD diagrams. Tom Ruen (talk) 20:58, 8 January 2010 (UTC)

Standard spaces
I think that most of the section on standard spaces, that has been recently added, is out of place in this article. It seems to have been copied from the other pages, and a one-liner with one or two links is all that is needed here. Also, no reference has been given for the term "standard space". Opinions? -- Cheers, Steelpillow (Talk) 21:06, 25 February 2010 (UTC)
 * Yes, it's not something I've seen before but thinking about it it's rather trivial: of course you can map e.g. a 7-dimensional simplex with 8 vertices to the cube in three dimensions, and as every two vertices are joined you also get every edge, every face etc.. With more complex polytopes even just in three dimensions you need ever higher dimensional and so more abstract simplexes; but the map is not one-to-one so you can't turn the polytope into a simplex. So it doesn't tell you anything about the polytope.-- JohnBlackburne wordsdeeds 21:23, 25 February 2010 (UTC)

Coxeter-Dynkin diagram graphics
If anyone feels able to contribute, please visit the discussion over SVG vs PNG formatting for these diagrams. We are trying to establish a consensus to end a reversion war, and there are literally hundreds of instances to sort out. 83.104.46.71 (talk) 19:05, 11 March 2010 (UTC)

Triacontakaiditeron
I think Triacontakaiditeron should redirect to 5-orthoplex. --84.61.182.248 (talk) 17:16, 22 December 2010 (UTC)

And Hexacontitetrapeton should redirect to 6-orthoplex. --84.61.182.248 (talk) 18:18, 22 December 2010 (UTC)

Naming polytopes in higher dimensions
Hi, I am the author of the note that Octahedron80 cited and Arthur Rubin felt is "not usable as a reference; appears to be the author's original research, without much indication of the author's identity or expertise".

First, I'd like to point out that wp:or applies to content not references - the whole point of many papers is to document the author's original research. That's why they get referenced - it's what references are for. Using wp:or to criticise such references is simply mistaken.

Second, my note acknowledges help from among others Prof. Norman Johnson (of Johnson solids fame among other things), and George Olshevsky who also needs no introduction.

My identity is simply exposed by following the link at the top of the note. Since the note is published on my website, I felt that confirmation enough. Ah, well, I guess that one man's obvious is another man's one-click-too-many-away.

My expertise must be left for others to judge, but again as evidence I offer my website, together with some of my peer-reviewed papers I reference and reproduce there.

I would not comment here, except that I feel the reversion of Octahedron80's well-meant edit should really have been discussed first - but then I have to declare an interest don't I ;-)

The edited page before reversion is here and its diff is here

— Cheers, Steelpillow (Talk) 21:46, 14 January 2011 (UTC)

 * Thanks Guy. Your writeup at is excellent for showing the origins and development of the terminology for dimension-dependent polytopes. The link to your page was also added to 5-polytope, and equally reverted there. However it was already given in a intro citation note  there as well. Anyway for discussion the central section should at 5-polytope which I wrote up originally. These names are used exclusively in the context of uniform polytopes, although polychora may be used more widely, like hendecachoron. I've mostly tried to keep most usage within uniform polytope terminology more generic. The simplex family is where it is most used as a suffix: pentachoron (5-faceted polychoron, same as 5-cell and 4-simplex), hexateron (6-faceted polyteron, same as 5-simplex), etc, but n-simplex can always be used. But I have no printed sources, and Norman's book on uniform polytopes is yet to be published. Tom Ruen (talk) 22:34, 14 January 2011 (UTC)

 * I tried to find the the name of the author of the site, and got a 404 after 3 clicks from the page heading where I thought information about the author should be, and that page doesn't attribute the names to George. He may need no introduction in the field, but he does need an a reliable source naming him for the attribution to be listed.   — Arthur Rubin  (talk) 01:34, 15 January 2011 (UTC)
 * Hey, thanks for alerting me to the 404! I'll fix that when I get a moment. Since my note is getting so much attention, I'll also see if I can tidy up some of the issues you raise. Hope nobody's in a hurry, I have some more urgent stuff to do first. Meanwhile I am puzzled why you have a problem with attributability to Olshevsky. My note contains a paragraph which explains all this. Are you perhaps concerned that I did not provide a reference? That was not possible - I was giving the first coherent published account of it. — Cheers, Steelpillow (Talk) 16:19, 15 January 2011 (UTC)

I don't know what you are talking about. I just have seen some example out there that used the words (2006) BEFORE here words were tagged (2010). "Original research" means that a Wikipedian newly creates something which has never been seen somewhere before; SO it is not in the case. I am pointing out that I just refer to the secondary source. If you don't accept this policy, delete those words as well (in every related articles also). --Octra Bond (talk) 09:37, 15 January 2011 (UTC)

 * To begin with, I apologize for not recognizing "Guy" as being sufficient (self-)identification as a recognized expert. (It should be pointed out that Bowers is apparently not recognized by the project as an appropriate expert, at least for naming conventions.)  Nonetheless, the article attributes two different naming systems, one to George, and one to Wendy, without indicating which is primary, although the table seems to be Wendy's.  If the reference is considered reliable, both naming systems should be included, where they differ.  — Arthur Rubin  (talk) 16:11, 15 January 2011 (UTC)
 * No need for an apology over my name - I never wrote the note with formal referencing in mind. You remark on two other unrelated issues: whether Bowers' system should be used on, and whether my note suffers shortcomings of its own. If you feel that it is time to embrace Bowers, that is a different issue to be pursued under its own topic. If you feel obliged to peer-review my note, that again my be pursued elsewhere - I would be happy to correspond. — Cheers, Steelpillow (Talk) 16:47, 15 January 2011 (UTC)
 * No, I don't feel your note has shortcomings, actually. However, Octohedron used it to source George's naming system, while you seem to prefer Wendy's, even though it doesn't go beyond 9-topes.  If your site is considered a valid reference by Wikipedia standards, we should include your preferred naming system.  — Arthur Rubin  (talk) 18:03, 15 January 2011 (UTC)
 * Didn't he use it to source Wendy's (as extended by me and used on )? Which reminds me I need to add polyyotton (or polyotton?) to my list, but I personally would get rid of polyxennon - "Xenna-" has not been ratified for SI and has several competitors, see Non-SI unit prefixes. When SI does get a 10 27 prefix, we will probably have to discard polyxennon anyway. — Cheers, Steelpillow (Talk) 19:00, 15 January 2011 (UTC)

 * p.s. Richard Klitzing's enumerations of the uniform polytopes includes up to 10D (polyxennon), and names up to 13D as polyudekon, following a reverse alphabetic sequence, similar to some proposed SI extentions. Tom Ruen (talk) 22:39, 15 January 2011 (UTC)

 * The earliest web reference to my names (teron to yotton) is, which is the 2003 version of the polygloss . I suggested the use of metric prefixes like zetto- to get around the apposition of two numbers (eg twenty four-D vs 24-D).  Metric prefixes are meant to be apposed to number.

 * The names were advanced and eventally adopted in a private list that a number of people in this list, such as Norman Johnson, George Olshevsky, and Guy all read. The polygloss was written one christmas day (2002), with the intent to simplify the revised notation that these form part of.

 * Since the names are advanced by others, rather than me, it is hardly original research, but rather a recommendation to use this rather than the more generic set of names. It's rather like the name of new elements, but we haven't set down to decide the 'winner' yet.   The PG does list George's names under, along with the full, but never used, thousand-rule scheme.  I have not seen Dr Klitzing's extensions past my naming, or even Guy Inchbald's referenced paper.  My recollection is that George's list derives from mine.

 * Choron is from GO *chemera (room), simplified by NWJ. I had no reason to alter this name, so it is assumed without modification.  However there were no names for the higher stuff.

 * W --Wendy.krieger (talk) 09:02, 16 January 2011 (UTC)

 * Thanks for the clarification, Wendy. It seems to me that the 2- to 9-dimensional names are constant across the various lists (mine is here), though there is the odd typo here and there. But the others are still in flux, so are best referred to as n-poyltopes, possibly with references to some names used by notable authors (and I think we need to be vary careful about who might be notable). I seem to recall someone introducing "polyxennon" just to tidy up the set of 10 dimensions deemed worthy of their own articles. But I find this artificial and unjustified by the facts. So: -

 * I have no idea why the 5-polytope article was left in such a state for so long, but I've done my best to correct it now. ~  Keiji (iNVERTED)  ( Talk )  17:13, 9 March 2011 (UTC)

Proposed demotion of "polyxennon"
Since the term polyxennon is not widely accepted and breaks the "SI prefix" convention that does seem to be becoming accepted, I propose that we move the Uniform polyxennon article to Uniform 10-polytope and confine "polyxennon" to redirects and in-page references.

 * Votes
 * For — Cheers, Steelpillow (Talk) 11:54, 16 January 2011 (UTC)
 * Support the move. I don't see why we even need to have in-page references to the non-widely-used name. —David Eppstein (talk) 17:06, 16 January 2011 (UTC)
 * I have no grounds for an objection. I did my best to set up the articles to not depend on any specific names. The uniform n-polytope listings were stopped at 10 simply as a round number. Originally it was under 10-polytope, but I moved because the contents were focused entirely on the uniform ones. Tom Ruen (talk) 23:40, 16 January 2011 (UTC)
 * No Comment I had technical difficulties with this, since all of x, e, n are in use elsewhere.  We have, from n=-1 to -8, -1n 0v 1e 2h 3c 4t 5p 6x 7z 8y.  It would be acceptable if one puts -1w (wessian = existance), then 9n xenno- would indeed work.  I suppose this would get around all those people who think of polytopes made of things as well.  Eventually, we're going to head faster than the SI prefix rule, in any case, So we need to address issues at hand.  --Wendy.krieger (talk) 07:06, 17 January 2011 (UTC)
 * You write as if we're free to make up notation here as a way of promoting it elsewhere. We're not — Wikipedia is a follower, not a leader. See WP:NOR and WP:SOAP. —David Eppstein (talk) 07:52, 17 January 2011 (UTC)
 * We're not actually making up new notation. Polyteron in my meaning is now the top six pages of google, except for a sole reference in spanish to the drug.  The particular debate here is whether we should accept 'xennon' from list B (Bowers-Olshevsky) into list A (Krieger).  Bowers-Olshevsky do follow my list as far as it goes, and then uses a different rule of number contraction to go higher.  An alternative would be to use the rule of Wilberforce Mann, which goes past this dimension, and also meant to stand with numbers.
 * Since the PG is as much about language use as it is about mathematics, one needs to consider the linguistic issues and clashes of meaning, when adopting new terminology. What I listed above is in the PG under 'chevn', 'thousand-rule' (on th/þ), as well 'polytope name'.  The chevn-rule is about heading up columns with letters representing dimensions, eg c3 h2 e1 v0 n-1, or c4 e3 h2 v1 n0, according to the rules of multiplication.
 * So my comment is more about what is needed to bring a list B name into list A. Note PG lists both A and B under thousand rule entries.  --Wendy.krieger (talk) 07:59, 18 January 2011 (UTC)

Well, having thought about this a while I think there is a clear mandate to expunge "polyxennon" from article titles and if need be section titles. The term may be used in the text where its origin needs to be cited. If any wathcer of this article wishes to contest this, please confine your arguments to Wikipedaia policy/guidelines and post below or on my talk page. — Cheers, Steelpillow (Talk) 18:52, 24 May 2011 (UTC)

 * I added a move request at Talk:Uniform polyxennon. Tom Ruen (talk) 19:44, 24 May 2011 (UTC)

polyteron, etc
Much of this new notation stems from my constructions over a decade ago, eg in the polygloss, and also in "Symmetry: Culture and Science" paper "Walls and Bridges: the view from six dimensions". While i doubt that 'google scholar' would show many results, the regular google search shows many results on this word. The whole point behind the polygloss was to make things easier for people to visualise the higher dimensions, by removing meanings of words that do not continue to have that force in the higher dimensions. In any case, the terminology has been adopted by other sites, for example, George Olshevsky.

--Wendy.krieger (talk) 07:23, 1 July 2011 (UTC)

 * Wendy, I wish this terminology could be recognized on wikipedia. I agree 'google scholar' ought not to be the singular online standard for geometric terminology. Norman Johnson suggested polychoron with George Olshevsky for 4-polytopes, but John Conway hasn't expressed anything. George Olshevsky has published information on the convex uniform polychorons, but nothing on his archived website for higher dimensional names, even if in private. Guy Inchbald has a good summary of the subject at, and Jonathan Bowers goes a bit wild with his fun attempt to systematic naming . We have your list at Polygloss which along with Jonathan, diverges from the SI prefixes to say polyectons for 7-polytopes. Who sets the standards and what prevents you from arbitrarily deciding next week to change the terminology again, and which version should wikipedia support (even assuming self-published sources were acceptable)? At least in the days of printed books, there was a fixed edition to reference, and even if corrections were later attempted, if usage takes off, authors lose control to "correct" the terminology they created and their first publishing. It's messy, but the real work I think must be to get active geometers to use the language and publish with it, and then we have something to cite. Tom Ruen (talk) 18:58, 1 July 2011 (UTC)

 * Have to support Tom's view here. Wikipedia is quite strict about this. Several of my ideas cannot be discussed here for the same reason. — Cheers, Steelpillow (Talk) 19:09, 1 July 2011 (UTC)

Facets
The disambig page Facet (disambiguation), recently had some material cleaned out, and moved to Facet (geometry). I've now moved some remaining material there.

One item that was remaining at the disambig page, was the interwiki link to de:Polytop (Geometrie) - but that seems to be aimed here. Possibly this will help someone with ideas of further work, or pages that need to be merged or integrated. –Quiddity (talk) 19:38, 8 June 2013 (UTC)

Body, facet, ridge
I notice Norman W. Johnson in his preprint Geometries and Transformations, defines the words scope, chamber, and wall in topological parallel to body, facet, and ridge respectively, in regards to n-honeycombs in n-space. I've not read previously on distinct terms for infinite vs finite forms. Tom Ruen (talk) 18:01, 27 February 2015 (UTC)
 * Just please don't start using them here until the wider mathematical community accepts his terminology. — Cheers, Steelpillow (Talk) 20:45, 27 February 2015 (UTC)
 * That's why I put it here, and perhaps these terms or others are referenced elsewhere as well. Tom Ruen (talk) 20:52, 27 February 2015 (UTC)

Who coined the term?
We have a link to the person who "translated" the original term by simply adding an -e, but none to the person who actually coined the term in the first place. Unfortunately, I couldn't find a reference, and the German article de:Polytop isn't helpful, either. Does anyone know who this "Hoppe" (de:Hoppe) was? It's a common name, and there are a number of mathematicians with that name. — Sebastian 16:52, 19 August 2015 (UTC)
 * According to Coxeter's Regular Polytopes (Dover Edn. p. 317), Reinhold Hoppe is the one you are after. The Internet seems to think his biog. dates are 1816–1900. HTH — Cheers, Steelpillow (Talk) 17:59, 19 August 2015 (UTC)
 * We have articles on him in de:Reinhold Hoppe and nl:Reinhold Hoppe. As an important journal editor and member of Leopoldina he's clearly notable, so we should have one here too. The Dutch article says he's also known for proving that someone else's solution to the kissing number problem (how many unit spheres can touch a central unit sphere in 3d) was bogus, and then publishing his own solution which also turned out (many years later) to be bogus. Neither article mentions coining "polytope" but Coxeter can be used to source that as well. —David Eppstein (talk) 19:08, 19 August 2015 (UTC)
 * Coxeter's note on his coining the term is on Page iv of the Dover edition. — Cheers, Steelpillow (Talk) 19:23, 19 August 2015 (UTC)
 * That's weird, my copy (the 1973 edition) doesn't have a page iv, and p.317 is just the index page that has Hoppe's name on it. The pages (144 & 165) that the index page links to do mention Hoppe, but in unrelated contexts. Instead the note on the coinage is on page vi.—David Eppstein (talk) 19:30, 19 August 2015 (UTC)
 * Ok, new article created. —David Eppstein (talk) 20:11, 19 August 2015 (UTC)
 * Wow, thanks a lot, that was quick! — Sebastian 20:33, 19 August 2015 (UTC)

Introduction picture
The picture in the introduction is the same as the picture for polygons. This can lead to confusion. These two pages have similar titles and talk about similar things. the polygon page also links here, and visa versa. A person could get easily confused and think they are on the wrong page.The introduction should really have a 3rd+ dimension picture. — Preceding unsigned comment added by IllQuill (talk • contribs) 00:58, 19 April 2016 (UTC)
 * An n-dimensional picture would be most appropriate though sadly impossible. But I agree that something better than a few polygons would be welcome. (By the way, we don't top-post here. A new discussion should go below the earlier ones.) — Cheers, Steelpillow (Talk) 08:06, 19 April 2016 (UTC)

Description
It seems the intro describes a polytope as a polytope. Quite clarifying. Madyno (talk) 09:37, 31 July 2017 (UTC)
 * Yes, recursion is powerful, although knowing when to stop is the funny part. Tom Ruen (talk) 15:08, 31 July 2017 (UTC)

title: Talk:Polytope