# Talk:Tetrahedron

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## ( Written before TOC )

Maybe I'm just spacially challenged, but I can't agree with the following sentence from the article:

A tetrahedron can be embedded inside a cube so that each vertex is a vertex of the cube, and each edge is a diagonal of one of the cube's faces.

If you obey "each vertex is a vertex of the cube", then the statement that "each edge is a diagonal of one of the cube's faces" is wrong, maximally three edges of the tetrahedron can be diagonals of the cube, the others must lie on the edges of the cube. WDYT? --snoyes 00:17 Feb 26, 2003 (UTC)

Consider vertices at (0, 0, 1), (0, 1, 0), (1, 0, 0) and (1, 1, 1) on the unit cube.

The edges of the tetrahedron made by these points are:

• (0, 0, 1), (0, 1, 0)
• (0, 1, 0), (1, 0, 0)
• (0, 0, 1), (1, 0, 0)
• (0, 0, 1), (1, 1, 1)
• (0, 1, 0), (1, 1, 1)
• (1, 0, 0), (1, 1, 1)

All face diagonals (I think: it's late, I'm tired). The Anome 00:35 Feb 26, 2003 (UTC)

Yip, I'm just spaTially (and spelling) impaired. Thanks for your time. --snoyes 01:24 Feb 26, 2003 (UTC)
The buckminster-fuller web site has a great animation that shows this :-) -- Tarquin 11:11 Feb 26, 2003 (UTC) ... and I can't find it on Google. I have it on my HD though, email me Snoyes and I'll send it back to you :-)

A cube can be divided into five tetrahedra. The central one is a regular tetrahedron whose edges are the diagonals of the cube's faces. The other four tetrahedra are not regular with three edges diagonals of the cube and the other threee edges edges of the cube. Thus everything said above is correct. You're just talking about different tetrahedra.

## Changed CTB to CTA in the volume formula

I have verified the changed formula is correct with a computer program, but not by mathematical proof. It would be good if the person who originally entered the formula could verify as well. Royappa 06:49, 1 January 2006 (UTC)

## New stat table

I replace stat table with template version, which uses tricky nested templates as a "database" which allows the same data to be reformatted into multiple locations and formats. See here for more details: User:Tomruen/polyhedron_db_testing

Tom Ruen 00:52, 4 March 2006 (UTC)

## edit of irregular tetrahedron symmetries

What has been here the last 6 months or more is the following:

An irregular tetrahedron (3-sided Pyramid (geometry)) with equilateral base and the top vertex above the center has 6 isometries, like an equilateral triangle.
A tetrahedron composed of two pairs of identical isosceles triangles is such that the two edges that adjoin identical triangles are opposite and perpendicular, and thus such a tetrahedron (or digonal disphenoid) has one twofold rotational axis passing through the centers of the two edges that adjoin identical triangles (in the case where all four triangles are identical and the figure is a tetragonal disphenoid this is also a fourfold improper rotational axis); there are also two mirror planes, each passing through one of these two edges and extending through the center of the opposite edge.
In other cases there is no rotational symmetry and at most one mirror plane.

This is inaccurate (according to this there are 3 or 4 cases, depending how you read it). I have put a more careful enumeration of the 7 possibilities. --Andrew Kepert 23:14, 10 August 2006 (UTC)

## Proof for volume of tetrahedron

As stated above, someone requested the proof for the volume formula. I did not originally post the formula, but a proof can be found here:

I also think we should include this formula for the volume of a tetrahedron (found on http://mathworld.wolfram.com/Tetrahedron.html):

If a,b,c, are the three polyhedron edge vectors from a given polyhedron vertex, the volume is:

${\displaystyle {\frac {1}{3}}\displaystyle |a\cdot |b\times c||}$

--CmaccompH89 19:05, 22 August 2006 (UTC)

That should be ${\displaystyle {\frac {1}{3!}}\displaystyle |a\cdot (b\times c)|}$ Tom Duff 21:48, 12 September 2006 (UTC)
In order to remove the citation request I have relocated the associated paragraph to show a progression from triple scalar representation of volume to the formula using only the three edge lengths and three angles at one vertex. Frank M Jackson (talk) 18:36, 20 March 2008 (UTC)

## law of sines

${\displaystyle \angle OAB\cdot \angle OBC\cdot \angle OCA=\angle OAC\cdot \angle OCB\cdot \angle OBA.\,}$

Multiplying the angles themselves is so unusual that it ought to be expressed also in words. And if this is not multiplying the angles, then its meaning is obscure enough that it ought to be expressed also in words. —Tamfang 05:32, 3 May 2007 (UTC)

## Setting d=0 by translating tetrahedron

Initially sentence is awkward: "If d = 0 is the origin of the coordinate system, then the formula becomes"

Simplification improves sentence, but removes insight into why we may set d=0: "If d = 0, then the formula becomes "

I have tried to keep clear sentence structure while retaining that insight (unfortunately making the sentence much more verbose): "If the tetrahedron has been translated so that vertex d is at the origin, then d = 0, and so"

Thoughts on the best way to present this? Unsigned comment added by 98.203.237.75 (talk) 14:44, 17 January 2008 (UTC)

I agree. However, I would rather describe it as a reference frame choice, than a translation of the tetrahedron. Please see my edit. Paolo.dL (talk) 16:36, 17 January 2008 (UTC)

Ok, sounds good! —Preceding unsigned comment added by 98.203.237.75 (talk) 17:44, 17 January 2008 (UTC)

## Solid angle and radii of spheres

I have added solid angle, radii of spheres associated with a regular tetrahedron and position of exsphere center into first table. Frank M Jackson (talk) 09:39, 19 March 2008 (UTC)

## Practical applications

I miss some words about the use of tetrahedron-shaped nail constructs suitable for stopping cars... because no matter how many times they fall, one peak will always point up. Eh, I am sure someone will find better words than I do. —Preceding unsigned comment added by 129.240.72.130 (talk) 10:22, 11 April 2008 (UTC)

## the general case

Had I written it, I'd prefer "general tetrahedron" to "generalized tetrahedron". Opinions? Rationale? —Tamfang (talk) 06:31, 27 May 2008 (UTC)

Yes, as far as I know, generalized has a different meaning. From Cambridge Dictionary:
generalized, UK USUALLY generalised
involving a lot of people, places or things:
He spoke of generalized corruption in the government.
Isolated showers will give way to more generalized rain later in the day.
NOTE: The opposite is localized.
The term is also used in statistics, meaning "(improperly) applied to a wider population". Paolo.dL (talk) 21:01, 28 May 2008 (UTC)

## Second paragraph

I removed from the second paragraph the claim that tetrahedrons are the "second most common" type of pyramid. Perhaps it meant the second most common pyramid shape used for building classical monuments, which belongs at pyramid and not here (and would need a citation), or the second most commonly taught pyramid shape at school, which seems fairly unimportant and quite debatable. In a mathematical context (which this article is), "most common" is completely meaningless.

I replaced it by pointing out that tetrahedrons are not mearly pyramids but specifically are triangular pyramids (which is obvious of course, but only if you know it) Quietbritishjim (talk) 16:36, 15 August 2008 (UTC)

Excellent edit! Wasn't necessary to explain it though :-) When I saw the diff it was a pretty obvious enhancement. Tomeasy T C 17:31, 15 August 2008 (UTC)

## controversial formulae

These are in Coxeter Regular Polytopes, Table I(i):

 dihedral angle circumradius midradius inradius surface volume acos(1/3) ${\displaystyle {\sqrt {\frac {3}{2}}}l}$ ${\displaystyle {\frac {l}{\sqrt {2}}}}$ ${\displaystyle {\frac {l}{\sqrt {6}}}}$ ${\displaystyle {\sqrt {3}}a^{2}}$ ${\displaystyle {\frac {\sqrt {2}}{12}}a^{3}}$

where l is half the edge length a. —Tamfang (talk) 19:53, 21 November 2008 (UTC)

What's your point? Supporting the current state of the article, objecting it, or providing a reference? Tomeasy T C 01:29, 22 November 2008 (UTC)
The last. —Tamfang (talk) 03:08, 22 November 2008 (UTC)
You could do some very nice thing then: Add the reference to the article. Tomeasy T C 10:42, 22 November 2008 (UTC)
Shall I help you adding the reference to the article? Tomeasy T C 18:08, 24 November 2008 (UTC)
Sorry, I have just realized that you had put the reference already. Well done! 15:50, 25 November 2008 (UTC)

## Newly discovered property of Regular Tetrahedron?

Some time ago I stumbled on a web site that states following: If a regular tetrahedron is intersected with a plane that is parallel to two opposite edges, anywhere between the edges, the resulting 2D shape is a rectangle (or square) with perimeter equal to twice the edge dimension. It is easily provable. I have not seen this in the literature about the tetrahedron. Is this a new theorem?-Drova (talk) 13:40, 4 February 2009 (UTC)

I read the article and I am convinced that the statement is true. I am a little bit reluctant to believe the claim that this property is only true for regular tetrahedra. This aside, I would welcome the inclusion of this property accorded by the visual proofs. Tomeasy T C 15:25, 4 February 2009 (UTC)
If you unfold a tet into a linear net of four triangles, this rectangle is the straight line that transverses the net parallel to the top and bottom. If your line is drawn midway between top and bottom you get a square. Cloudswrest (talk)
An easy way to visualize this rectangle is to set the tetrahedron vertically on one of its edges. Assume the bottom edge goes left/right. The opposite top edge is perpendicular, also level with the ground and goes from front to back. Now slice the tet with a horizontal plane. The intersection is this rectangle. If the plane is nearer the ground the left/right edges will be the longer and the front/back edges will be the shorter. As the plane rises vertically the edges change and reverse rolls. At the midpoint in height you have a square. This square can be thought of as the "equator" of the tetrahedron. It traverses ALL faces of the tet similarly without any turning other than that induced by traversing the dihedral angle between faces. The tetrahedron is the only regular polyhedron that admits a discrete Hopf fibration in three dimensions. This is because this is a degenerate case with only ONE ring. A second or more rings would collide or intersect. Cloudswrest (talk) 03:48, 8 October 2015 (UTC)

Unfortunately, I have never contributed images to Wikipedia before. Can files in PDF or JPEG formats be used for such purpose. It would be better if some one with experience do that. It appears to me that tetrahedra with sides that are not equilateral triangles would be excluded from this rule, however I am not convinced either.Drova (talk) 16:44, 4 February 2009 (UTC)

Well, if you think of tetrahedra only, then a non-regular tetrahedra does not even have a unique edge dimension; thus, the statement does not apply by conception. However, one might think of complete different geometries, but I am not enough an expert to say something about this. Important for us is that the statement is a property of the regular tetrahedron.
If you like, you can add a statement about the property in the article without illustration. Others would probably refine the wording. However, the addition of an illustration would greatly support understanding the statement. Tomeasy T C 17:18, 4 February 2009 (UTC)
I disagree. I believe the statement must have been long known, but I think it should only be added to the article if appropriate sources can be found; otherwise, we're committing original research. Incidentally, the same property is true for any tetragonal disphenoid. —David Eppstein (talk) 17:52, 4 February 2009 (UTC)
As already mentioned, the statement does not apply to a disphenoid, because it does not have a unique edge length. However, one might reformulate and say that the perimeter of the rectangle is equal to twice the length of the opposing edges, which you probably meant. So, fine it is true for these tetrahedra as well. I think I even have a clear picture on how one can state this for an arbitrary tetraheadron. (The perimeter of the rectangle would be twice an interpolation of the lengths of the two opposing edges. The interpolation depends linearly on the distance of the plane and the respective opposing edge). Anyways... The original statement is still valid for the subject of the present article and worth to mention. Especially when illustrated, I find it adds value.
Personally, I do not see the OR problem as long as the derivation is purely logical and commonly comprehensible. However, I might be wrong and tried to see whether WP:OR says something specific about mathematical derivations, but it does not. So, I started a talk page section there, because I the question as to when a logical derivation becomes OR is anyway interesting. Tomeasy T C 18:59, 4 February 2009 (UTC)

If the statement has been long known, we should be able to find an appropriate source that we could refer to. So far, I haven't been able to find one. I think it is an interesting statement, that would enhance the article. It is hard to believe that it is completely novel. In regard to tetragonal disphenoid, the Wiki article referenced above states that a regular tetrahedron is not considered to be a disphenoid.Drova (talk) 18:32, 4 February 2009 (UTC)

The disphenoid tetrahedron graphic I made, showing this tetrahedron inside of a cuboid shows clearly that cross sections from this orientation (along the y axis) will be rectangles. So this appears to be true at least for all digonal disphenoids, tetrahedra with 4 equal edges and two opposite (not necessarily) unequal edges. SockPuppetForTomruen (talk) 20:04, 4 February 2009 (UTC)

Perhaps a person with good imagination can see rectangles in the above disphenoid. Such person would also be able to imagine rectangles in almost any image of a tetrahedron. What interesting about the statement is that all such rectangular cross sections have constant perimeter equal to twice the edge distance. That can be easily proven for regular tetrahedron with edge E. If distance from the vertex to the point of intersection of the plane with the edge (measured along the edge) is a, then, from equilateral triangles, resulting rectangle would be (a)x(E-a). Perimeter of such rectangle is 2(E-a+a)=2E. Is there another proof, that is not based on equilateral triangles?-Drova (talk) 23:00, 4 February 2009 (UTC)

Here's an external web site that illustrates this rather well http://www.matematicasvisuales.com/english/html/geometry/space/sectetra.html Cloudswrest (talk) 19:34, 8 October 2015 (UTC)

## Regular tetrahedron coords?

I feel this would be useful addition to the article. It certainly would have helped me. I'm not sure where to put them however:

x y z
A 0 0 0
B 1 0 0
C 0.5 sqrt(3)/2 0
D 0.5 sqrt(3)/6 sqrt(2/3)

--Skytopia (talk) 00:01, 16 February 2009 (UTC)

Or even simpler, alternated vertices of a cube! SockPuppetForTomruen (talk) 03:06, 16 February 2009 (UTC)

x y z
A 1 1 1
B 1 -1 -1
C -1 -1 1
D -1 1 -1
Or indeed... that one. I'm amazed how those originals numbers can be simplified to ones and minus ones like that. Good stuff. I'm guessing there's not much point to having both coord sets in the article then? --Skytopia (talk) 14:27, 16 February 2009 (UTC)

Both are useful. The first comes from a dimensional series of regular simplexes. The second it just based on a special relation in 3D where the vertices of a cube contain two tetahedra. SockPuppetForTomruen (talk) 20:21, 16 February 2009 (UTC)

I added a section but noticed it is repeated under geometric relations. It is sort of a messy article, mixing regular and general tetrahedra. SockPuppetForTomruen (talk) 20:31, 16 February 2009 (UTC)

Okay, I'll leave it up to you and others here about adding the set I suggested as well. In any case, I like the addition you made to the start of the article - nice for them to get a mention early on.--Skytopia (talk) 05:53, 26 February 2009 (UTC)

## One common formula for circumsphere, midsphere and insphere radius

I suggest in addition to formulas of circumsphere, midsphere and insphere radius of regular tetrahedron to add one simple common formula that covers all three of them and is applicable even in dimensions other than 3. The formula is: ${\displaystyle R={\sqrt {n-k \over 2nk}}\,a\,}$

where n is number of points at equal distance to each other (for regular triangle n=3, for regular tetrahedron n=4, etc.)

and k is number of points defining an entity the sphere “touches” (for vertex k=1, for edge k=2, for face k=3 and so on).

This is my first post to Wikipedia and I'm not native English speaker so forgive me if something is wrong :)

Coxeter Regular Polytopes Table I(iii) gives
${\displaystyle R_{j}/l={\sqrt {{\frac {2}{j+1}}-{\frac {2}{n+1}}}}}$
where j is the dimension of the tangent element (0 for a vertex...), n is the dimension of the space, and l is the half-length of an edge. That's a match. —Tamfang (talk) 02:43, 22 June 2009 (UTC)
I suggest that you say that the radii of the circumsphere, midsphere and insphere for a regular tetrahedron are all related by the formula
${\displaystyle R_{k}={\sqrt {\frac {3-k}{8(k+1)}}}}$
where k is dimension of the entity that the sphere “touches” (e.g. for vertices k=0, for edges k=1, for faces k=2).
Then make a reference to Coxeter formula saying that it can be generalized for higher dimensional Simplices and insert the Coxeter formula into the section on Simplices. Hope this helps. Frank M Jackson (talk) 18:07, 22 June 2009 (UTC)

Hi Tamfang

I didn’t know about Coxeter formula, I delivered it myself:) But I think already ancient Greeks knew it, any way to derive it one needs only Pythagoras theorem.

Hi Frank M Jackson

I’m fully aware that my definition of n and k are somehow weird, but defining them this way makes the formula extremely easy to remember - take 1/k-1/n divide by 2 and take square root (2 comes from the fact that unit n-simplex has edge sqrt(2)).

I understand that section Formulas for regular tetrahedron is quick reference, so this simple formula can be easily there, since IMHO it is easy to remember and useful (replaces three formulas that are more dificult to remember). As for yuor sugestion to include it in Simplices I'm not so sure. If one reads about simplices than he/she problably can get this formlula on the fly, so why it should be there?

Firstly it would be useful if you set yourself up with a proper user account rather than remain anonymous. This enables easier discussion between users and you can get help from Wikipedia administrators.
I believe that the table at the front of the article is a quick and useful reference and should not be altered. The fact that the circumsphere, midsphere and insphere are all related should be mentioned but in a separate place within the article. Quote the published formula as advised by Tamfang rather than your own interpretation and make a reference back to Coxeter. It avoids confusion.
The Coxeter formula relates to the more general Simplex so it may be worth a mention in the Wikipedia article on simplices as there is no reference to it there, at present. Frank M Jackson (talk) 18:44, 24 June 2009 (UTC)
I'd adjust the Coxeter formula to use the full edge length, conforming to the rest of the table.
${\displaystyle R_{j}/a={\sqrt {{\frac {1}{2(j+1)}}-{\frac {1}{2(n+1)}}}}}$
(someone check my algebra). —Tamfang (talk) 04:05, 29 June 2009 (UTC)

## Minimum Volume property

It is said that a regular tetrahedron is the solid with the minimum volume for a given surface area, while a sphere has the maximum. See ‘Tetrahedral Hypothesis’, an outmoded geological speculation. This seems quite an important property, and maybe one of you clever chaps could say something about it? John Wheater (talk) 07:38, 1 July 2009 (UTC)

It's false for general solids, and even for general convex uniform polyhedra. Consider a uniform n-gonal prism (whose edges are equal): as n is increased with the surface area held constant, clearly the volume can be as small as you like. —Tamfang (talk) 17:06, 7 July 2009 (UTC)
Please don't go changing my spellings. I meant polyhedra (noun), not polyhedral (adjective). —Tamfang (talk) 20:09, 13 July 2009 (UTC)
I (no mathematician) am guessing that such ‘n-gonal prisms’ form a monotonically increasing series of forms, starting with tetrahedron and cube as 3-gonal and 6-gonal. Also that as n tends to infinity the form tends to that of a sphere. Now, for tetrahedron, cube, and sphere, we may express the volume as k*A*sqrt(A), where k is constant for each form and A is the surface area, (and I guess this true of the other members?). For tetrahedron, cube, and sphere, the constants are, I think: <sqrt(2)/(36*sqrt(3))>, <1/(6*sqrt(6))>, and <1/(6*sqrt(pi))>. These evaluate to about 0.0227, 0.0680, and 0.0940. So it looks to me that there is a flaw in your reasoning, and that Holmes was right in his 1944 geology book. What do you think?, and could something be said? John Wheater (talk) 06:46, 9 July 2009 (UTC)
Tamfang is correct, and you are not. Consider the rectangular boxes (a very special case of the prisms mentioned by Tamfang) with side lengths x, 1/x, 1/x. As x goes to infinity, the surface area will be 4 + 2/x2, very close to 4. However the volume will be 1/x, very close to 0. By choosing x large enough you can make the volume far smaller than that of a regular tetrahedron with the same surface area. —David Eppstein (talk) 07:09, 9 July 2009 (UTC)
If we restrict ourselves to the Platonic solid then the tetrahedron is the one out of the five with min volume, but thats not a particularly amazing result. The Tetrahedal Hypothesis may be worth an article [1]--Salix (talk): 08:04, 9 July 2009 (UTC)
I've created said article at Tetrahedal hypothesis.--Salix (talk): 16:05, 9 July 2009 (UTC)
Interesting, but shouldn't it be Tetrahedral Hypothesis or Tetrahedral hypothesis? Tom Ruen (talk) 19:19, 9 July 2009 (UTC)
Now at Tetrahedral hypothesis. —David Eppstein (talk) 20:22, 9 July 2009 (UTC)
David, what you say above is true, but I’m sorry to have caused you to waste your time pointing out the obvious fact that you can put two rectangles arbitrarily close together! I should have said that I was considering solids formed of polygons whose sides were equal, and, in my ignorance, I thought this was what Tamfang was discussing, and what I was exemplifying. It seems though (thanks Salix) that not all such polygons can fit together in the right way (perils of extrapolating the soccer-ball pentagon!). But is there some set of solids that includes the Platonics and the sphere? Holmes says “Since, for a given surface area, a sphere is the regular figure with the greatest volume, while the tetrahedron is the regular figure with the smallest volume, it was thought that a contracting globe would tend to shrink to a tetrahedral form”. And Qazi (2004), cited in Tetrahedral hypothesis, says this is a “basic principle of geometry”. If there is a set of Volume/Area ratios for “regular figures”, and the sphere and tetrahedron are the end-points, that does seem interesting to me, but I’ll pipe down now if nobody else thinks so. The Tetrahedral hypothesis itself is nothing but a historical curiosity, abandoned long before plate tectonics was accepted, but the idea of a sphere “contracting” to a tetrahedron is intriguing.
In this context "regular figure" must mean Platonic solid. Tamfang's example of a prism with all faces being regular polygons has much smaller volume, for a given surface area, than a regular tetrahedron. —David Eppstein (talk) 18:14, 10 July 2009 (UTC)
Yes, thanks, good. Now I know the meaning of prism and Platonic solid I at last understand: all the faces are regular polygons, but not the same regular polygon! Tamfang, like you, was provoked into pointing out that two plates of arbitrarily large area can be placed close together! If only I’d said ‘regular solid’…I note that a sphere is not actually a Platonic solid, but I guess an icosahedron is close, with a volume/area constant of 0.0856 (cf. tetrahedron 0.0227 and sphere .0940). So this article is not the place for the ‘minimum volume’ property; I might try adding something to Platonic solids and Tetrahedral hypothesis. Many thanks to all for a marvellous bit of education, and goodbye (at last). John Wheater (talk) 10:17, 11 July 2009 (UTC)
"But is there some set of solids that includes the Platonics and the sphere?" Well, you can make a set of anythings, heh. I guess you'd be most interested in the family of "geodesic" spheres, which approach a sphere as the number of pieces increases. —Tamfang (talk) 20:15, 13 July 2009 (UTC)

## Tetrahedron is also a snub digonal dihedron/hosohedron

{3, 3} is also s{2, 2} I believe? Se Wythoff symbol. Shouldn't this be in the article? --116.14.72.74 (talk) 03:10, 29 July 2009 (UTC)

Yes! It's in the table on the right. Tom Ruen (talk) 03:46, 29 July 2009 (UTC)

## convexity

The tetrahedron is the only convex polyhedron that has four faces.

Is there a non-convex polyhedron with four faces? —Tamfang (talk) 20:03, 15 August 2009 (UTC)

No, see the reference. Double sharp (talk) 05:36, 16 August 2009 (UTC)
This leads to a question about the smallest number of faces for a non-convex polyhedron. I can construct one with five faces, one of which is non convex and one with six faces all of which are convex. Not sure if its possible to construct a non-convex polyhedron with five convex faces.--Salix (talk): 09:21, 16 August 2009 (UTC)
I think not, but I haven't got a proof yet, and even if I did it would be original research. Not sure if there's a reference anywhere. I may do some Googling. Professor M. Fiendish 06:06, 22 August 2009 (UTC)
Can't seem to find it. Professor M. Fiendish 06:15, 22 August 2009 (UTC)

## Dual tetrahedron

This is my first Wikipedia edit so forgive me if this is not formatted correctly. There is a mistake near the top of this article where it says: If the original tetrahedron has edge length 1, its dual tetrahedron will also have edge length 1. I build polyhedral models and know that the Tetrahedron of edge length 1 will have a dual tetrahedron with edge length of 1/2. I could try a mathmatical proof but a simple model will show you that the 1 to 1 claim is incorrect. Please let me know if you implement this change. eric.higley@kla-tencor.com Thanks! Erichigley (talk) 16:13, 27 October 2009 (UTC)

Actually, the statement is misleading because the exact proportions of the dual depends on how you carry out the operation (i.e., the radius of the sphere of reciprocation). It is certainly possible to reciprocate a tetrahedron in such a way that its dual also has the same edge length, but this is not the only way to do it, as your models prove. This statement should probably be edited to not make these kinds of hidden assumptions.—Tetracube (talk) 17:06, 27 October 2009 (UTC)
OK I've now removed the line. TRhere might be a case for a more extensive treatment of duals lower down. --Salix (talk): 17:41, 27 October 2009 (UTC)
The hidden assumption, I think, is that the edges of the two polyhedra intersect; in other words, that the midspheres are equal. (See also canonical polyhedron. (Oh, darn.)) Do the other reg.poly. articles have analogous statements? —Tamfang (talk) 05:30, 12 November 2009 (UTC)

Actually, the dual of a tetrahedron is congruent to the original when inscribed in the same sphere. —The Doctahedron, 68.173.113.106 (talk) 22:44, 22 November 2011 (UTC)

Sure, but being inscribed in a sphere is not a self-dual statement: if a polyhedron is inscribed in a sphere, one would expect its dual to be circumscribed about the sphere. More specifically the polar reciprocation of a polyhedron inscribed in a sphere, with the polarity being defined with respect to the same sphere, gives a polyhedron circumscribed about the sphere. In contrast, the property of having a midsphere is self-dual: the polar of a polyhedron with a midsphere is a dual polyhedron with the same midsphere. For the other regular polyhedra, the edge length of the dual will be different than the edge length of the primal. —David Eppstein (talk) 22:53, 22 November 2011 (UTC)

## dimensions

User:Quietbritishjim asserts, *generalized* tetrahedra (i.e. not 3D) have non-3D properties (and thus the heading "Three-dimensional properties of a generalized tetrahedron" is not redundant). Does this mean other simplex articles ought to have sections such as "Two-dimensional properties of general triangles" and "One-dimensional properties of general line-segments"? —Tamfang (talk) 15:45, 13 November 2009 (UTC)

Sorry I didn't reply to this at the time, I've only just seen this comment. I actually don't know anything about this section, except for the comment that you've quoted. Someone has now removed "3D" from the title again, possibly making the same mistake, possibly because the section in fact also contains non-3D facts -- I don't have time to look properly at the moment. One thing I can say though is why tetrahedra are particularly interesting when generalised: they are one of I think only three platonic solids that generalise to an arbitrary number of dimensions. It's easy to construct the next one up, by just adding a point in the new dimension on the normal from the centre of the previous polytope. I've no idea why three dimensional properties of such a body would be particularly interesting though! Quietbritishjim (talk) 00:14, 11 February 2010 (UTC)
A tetrahedron is a 3D figure; a general tetrahedron is not an N-dimensional simplex but a 3D solid bounded by four triangles which are not necessarily equilateral. What "non-3D facts" do you see here? —Tamfang (talk) 02:55, 16 February 2010 (UTC)

## Tetrahedral Hypothesis

Thanks Salix for starting this article (see <Minimum Volume>, above), and to all who helped my understanding. I've expanded it a bit, and hope you sympathise with the new presentation - see discussion there. --John Wheater (talk) 11:43, 10 February 2010 (UTC)

## Dihedral Angle

I think that the dihedral angle is ,in fact, equal to arcsin{sqrt{3} / 3} , and arctan{sqrt{6} / 2} Would someone who knows how to edit that do so, under formulas, and the table? (A.glasswater (talk) 03:35, 10 August 2010 (UTC))

I've checked the formula and the ones in the article ${\displaystyle \arccos \left({1 \over 3}\right)=\arctan(2{\sqrt {2}})\,}$ are correct. You can see this by considering a tetrahedron of edge length 2. The dihedral angle can be calculated from a triangle formed from the mid point of one edge and the opposite edge, this has sides 2, ${\displaystyle {\sqrt {3}}}$, ${\displaystyle {\sqrt {3}}}$. From the Law of cosines the we obtain ${\displaystyle \cos \theta =1/3}$.--Salix (talk): 06:41, 10 August 2010 (UTC)

This is an application of the so called Law of cosines for a trihedral angle. In fact if we call V,A,B,C the four points of a regular tetrahedron where A,B,C are the points of the base, given a good orientation, we have that <AVB = <BVC = <CVA = pi/3 since the tetrahedron is regular. Now the law of sines states cos(<AVB) = cos(<BVC)*cos(<CVA)+sin(<BVC)sin(<CVA)cos(a) where a is the dihedral angle of the opposite face <AVB. Since all angles are equal cos(pi/3)=cos(pi/3)^2+sin(pi/3)^2*cos(a) cos(a)=1/3 a=arcccos(1/3)

Lhrrwcc (talk) 19:47, 16 February 2011 (UTC)

## Brain-numbing statements of syllogisms and worse

In the first paragraph:

"The tetrahedron is the only convex polyhedron that has four faces."
A). It says that a tetrahedron has four faces. That is the definition of "tetrahedron".
B). It is impossible to make any nonconvex polyhedron with just four faces.
Therefore all all tetahedrons are automatically convex polyhedrons.
If you assemble three suitable triangles into a pseudopolyhedron, then the size and the location of the fourth side are uniquely predetermined. It is impossible to place that fourth side to make the tetrahedron nonconvex. Therefore, stating that a tetrahedron is convex is a triviality. 98.67.97.108 (talk) 07:29, 13 August 2010 (UTC)

True. Meanwhile, please put new topics at the bottom, and don't correct other people's signed writings; they may have different style preferences. —Tamfang (talk) 19:04, 14 August 2010 (UTC)

## Formula and Notation Ambiguity

It seems to me that there's a fair amount of ambiguity in the formulae given in this article. For example, the formula for the radius:

${\displaystyle R={\frac {|\mathbf {a^{2}} (\mathbf {b} \times \mathbf {c} )+\mathbf {b^{2}} (\mathbf {c} \times \mathbf {a} )+\mathbf {c^{2}} (\mathbf {a} \times \mathbf {b} )|}{12V}}}$

uses the absolute value symbols to mean the length of a vector, while the formula for V uses the absolute value symbols to mean absolute value:

${\displaystyle 6V=|\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )|}$

Additionally, the notation ${\displaystyle \mathbf {x^{2}} }$ to notate the dot product of a vector with itself, as in ${\displaystyle \mathbf {a^{2}} (\mathbf {b} \times \mathbf {c} )}$, is potentially ambiguous to many readers, especially when the proper notation is ${\displaystyle \mathbf {x} \cdot \mathbf {x} }$ or ${\displaystyle \left\|\mathbf {x} \right\|^{2}}$. In some texts and many programming languages, ${\displaystyle \mathbf {x^{2}} }$ is the vector whose individual values have been squared. nben (talk) 17:28, 8 September 2010 (UTC)

This overloading of |·| doesn't bother me; but I agree that a² is unfortunate. (In geometric algebra it means something distinctly other than a·a.) —Tamfang (talk) 00:09, 11 September 2010 (UTC)
Changing   a²   to   a·a   looks messy and could also confuse. I suggest that a simple comment is inserted before the start of the formulae in this section that says − " Nota bene, in the formulas below, the scalar   a²   represents the inner vector product   a·a " − Frank M Jackson (talk) 14:02, 11 September 2010 (UTC)
Now implemented.Frank M Jackson (talk) 13:00, 13 September 2010 (UTC)
Works for me. The absolute value overload does not bother me that much, except that the double-line notation is used in the absolute value article when talking about vector spaces. I worry that readers will believe that the notation implies that the expression inside the absolute value brackets must ultimately be a scalar.
That said, I don't believe that in the formula for 6V it is correct to have the absolute value brackets at all, at least not when it is being applied to all formulas. E.g. If we use the expression for O, the circumcenter, in the article with the expression for V that I noted above (also in the article), we run into occasional problems due to the absolute value. If,
${\displaystyle \mathbf {a} =(-1,0,0),\;\mathbf {b} =(0,-1,0),\;\mathbf {c} =(0,0,-1)}$,
then the circumcenter is
${\displaystyle {\frac {\mathbf {a} ^{2}(\mathbf {b} \times \mathbf {c} )+\mathbf {b} ^{2}(\mathbf {c} \times \mathbf {a} )+\mathbf {c} ^{2}(\mathbf {a} \times \mathbf {b} )}{2|\mathbf {a} \cdot (\mathbf {b} \times \mathbf {c} )|}}={\frac {(1,1,1)}{2}}=(1/2,1/2,1/2)}$.
This is the same result we get if we use a, b, and c equal to the euclidean bases instead of their opposites. If the absolute value is removed from the formula for V, we get the correct result of (-1/2, -1/2, -1/2). A proper derivation is long but results in the same conclusions. Of course, removing the absolute value invalidates the expression in the article for the radius, so either that needs to be modified or the equation for O needs to not use V. nben (talk) 15:43, 14 September 2010 (UTC)
Well spotted. I agree that the vector equations should not have volume repesented in absolute value. I will make the change accordingly.Frank M Jackson (talk) 16:31, 14 September 2010 (UTC)

## Proof

Does Wikipedia have a proof that a tetrahedron cannot tile space?? The following proves a pentagon cannot tile a plane:

A regular pentagon has angles of 108 degrees. 360 / 108 = 3 1/3, which is not an integer, so a pentagon cannot tile a plane.

Any proof that a tetrahedron cannot tile space?? Georgia guy (talk) 18:45, 28 February 2011 (UTC)

Well, the dihedral angle of a regular tetrahedron is not a rational fraction of a circle. Many irregular tetrahedra do tile space, though. (And regular tetrahedra can tile curved space, but I don't suppose that's what you were after.) —Tamfang (talk) 22:14, 28 February 2011 (UTC)
Agreed. dihedral angle is 70.53°, not a divisor of 360, so something else (like 2 tets+2 octs) are needed. And for irregular-tet example: disphenoid tetrahedral honeycomb Tom Ruen (talk) 10:01, 1 March 2011 (UTC)

## A link to similar objects

I am wondering the name for the similar object taking (0,0,0), (0,1,0), (0,0,1), (1,0,0)??? 174.91.223.216 (talk) 14:56, 23 March 2011 (UTC)

That's an irregular tetrahedron; it could be called a right isosceles tetrahedron. —Tamfang (talk) 17:53, 23 March 2011 (UTC)
Do any scalene tetrahedrons exist?? (This means no 4 faces are congruent.) Georgia guy (talk) 17:55, 23 March 2011 (UTC)

You might want to take a look at Schläfli orthoscheme. That's on a different type of tetrahedron (with vertices (0,0,0), (0,0,1), (0,1,1), (1,1,1)) but it discusses work of Fiedler and Coxeter on triangles like the one you mention in which there is a spanning tree of pairwise perpendicular edges. As for whether there are tetrahedra with no pair of congruent faces: yes. If you pick four random points (in any reasonable continuous probability distribution) then with probability one it will have this property. —David Eppstein (talk) 17:58, 23 March 2011 (UTC)

## Degenerate form of antiprism and trapezohedron

Restored previous edit mentioning that a tetrahedron is a degenerate form of antiprism and of trapezohedron. These are not just random degenerate forms as implied by Steelpillow (talk · contribs), but are special cases in these families of polyhedra where n = 2. These also highlight relationships of tetrahedra to octahedra and hexahedra respectively. 75.150.168.6 (talk) 04:33, 6 October 2013 (UTC)

I trust that TomRuen's subsequent edits have found a better home than the article lead for those thoughts. — Cheers, Steelpillow (Talk) 09:18, 6 October 2013 (UTC)

## Incorrect definition of disphenoid

The page on disphenoids is inconsistent. Someone has been changing things to insist that a "digonal disphenoid" (two pairs of isoceles triangle faces) is a special case of a disphenoid (4 congruent faces). This is not true. Digonal disphenoids do not have 4 congruent faces in general. Here a disphenoid is incorrectly defined as a digonal disphenoid. 50.177.1.160 (talk) 21:29, 13 December 2013 (UTC)

How do you define a digonal disphenoid? The article says it has two pairs of congruent triangles. Some definitions say a disphenoid has 4 congruent triangles, but digonal diphenoid clearly contradicts that, so are you saying a digonal disphenoid is misnamed or misdefined? Tom Ruen (talk) 01:20, 14 December 2013 (UTC)
Linguistically, the two definitions are consistent. Consider for example elongating a cube parallel to one set of edges. The resulting figure is a cuboid, specifically an elongated cuboid: we add "-oid" to the root of "cube". Applying the same language rule to the sphenoid we would get a "digonal sphenoidoid" which is grammatically nonsensical - it is ether "like a wedge" or it is not, to say that it is "like a thing like a wedge" is superfluous. Linguistically we already have the "-oid" so there is no need to add it again. Thus, we find "digonal sphenoid" to be the correct form. What would probably be useful to help avoid this sort of ambiguity would be an accepted qualifier for the 4-congruent disphenoid, "equiangular disphenoid" or "equisphenal disphenoid," something like that, and a more general definition for the term "disphenoid". But it is likely that the mathematical community are in a mess over this sort of thing, in which case we can do no better than mirror that mess. — Cheers, Steelpillow (Talk) 13:12, 14 December 2013 (UTC)

## Isosceles Tetrahedron

I notice that the definition given here for an isosceles tetrahedron is different to that at MathWorld. Here it says their faces are isosceles triangles, but MathWorld says it means that opposite edges have the same length. The article Disphenoid seems to agree with MathWorld. Auximines (talk) 16:54, 22 January 2014 (UTC)

I would be astonished if mathworld were correct. Clearly the property of equal-length opposite pairs applies to a tetrahedron having four congruent isosceles faces. But it is not exclusive because it applies to any tetrahedron having four congruent faces, even if they are scalene. I would not expect such a scalene-faced figure to be correctly described as an "isosceles tetrahedron." This is not the first suspicious material I have seen on Mathworld. Stay cool and try to reference a more reliable source. — Cheers, Steelpillow (Talk) 17:51, 22 January 2014 (UTC)
It looks like the equal-opposite edge definition is well-used, like this reference [2] from John Leech (mathematician), which would define both the tetragonal and rhombic disphenoids as isosceles tetrahedra. Tom Ruen (talk) 23:04, 22 January 2014 (UTC)
Ah, of course, just because the tetrahedron is isosceles, it does not mean the face triangles are: in the tetrahedron the equal edges are not adjacent. Silly me. But I would now question whether an isosceles tetrahedron and disphenoid are synonyms. Is an isosceles tetrahedron with scalene faces really a disphenoid? — Cheers, Steelpillow (Talk) 10:34, 23 January 2014 (UTC)

Really? --JWB (talk) 21:42, 16 August 2014 (UTC)

According to Coxeter (Regular Polytopes, Table I) and a bit of algebra, it's 3. —Tamfang (talk) 02:02, 17 August 2014 (UTC)
And the article does not say anywhere that it is 8. The section Formulas for a regular tetrahedron gives formulas from Coxeter for both radii, and their ratio is clearly 3, not 8. So there is no problem here. Dirac66 (talk) 02:07, 17 August 2014 (UTC)
It is indeed exactly three. Simply dissect a tetrahedron into tetrahedra and octahedra of one-third side length. The circumsphere of the innermost tetrahedron is just the insphere of the original. — Cheers, Steelpillow (Talk) 10:50, 17 August 2014 (UTC)

## Ordinary

The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has four faces.

What does "ordinary" mean here? And is "convex" necessary? —Tamfang (talk) 03:06, 17 August 2014 (UTC)

"Ordinary" means that we are not discussing twisted non-Euclidean forms such as the hemicube, dihedra and so forth. One could write "Euclidean" instead, but I think the meaning is clearer this way - calling them Euclidean begs the question as to what is so significant about Euclid and we will have visitors whose maths is not even that advanced. "Convex" is not necessary as long as we keep "ordinary" or "Euclidean", although it is true, and I think it could be removed. — Cheers, Steelpillow (Talk) 10:59, 17 August 2014 (UTC)
Then I thought, we could just say "elementary". To a lay person that implies basic junior school stuff, to a mathematician it implies a derivation from the principles in Euclid's Elements. Both are true. But is such a play on words cheating? — Cheers, Steelpillow (Talk) 11:03, 17 August 2014 (UTC)
Since the exact meaning of ordinary (or Euclidean or elementary) will not be obvious to all readers, it should be defined on the article page. Perhaps in a footnote at the end of the sentence which uses it. Dirac66 (talk) 13:22, 17 August 2014 (UTC)
"Ordinary" is not a defined mathematical term in this context. It is a word used in expressing definite ideas, just like the words "exact", "meaning" and "obvious" which you used in expressing your definite ideas. That is why I prefer it to "Euclidean" in the present context. — Cheers, Steelpillow (Talk) 16:21, 17 August 2014 (UTC)

## Meaning of demicube?

I clicked demicube in the "Fundamental convex regular and uniform polytopes in dimensions 2–10" box at the bottom of a page and was redirected here. There doesn't seem to be any explanation of what "demicube" means on this page. - Rainwarrior (talk) 09:41, 21 April 2016 (UTC)

See demihypercube for the n-dimensional case. When n < 5 it's not that interesting because it coincides with some of the other families (it's the digon in 2D, the tetrahedron in 3D, and the 16-cell in 4D). Double sharp (talk) 12:35, 21 April 2016 (UTC)
I included the term in the section with alternated cube, and relinked demicube to that section. Tom Ruen (talk) 13:22, 21 April 2016 (UTC)

## 3-simplex?

Surely the lead paragraph should mention the other synonym of the tetrahedron, that it's a 3-simplex (that is to say, the 3-dimensional analogue of a triangle)? I'm sure at least one good citation for this can be found, on Wolfram Mathematics or a similar site.

(Does Wikipedia have a "simplex" article? Generally, a "N-[dimensional] simplex" is the simplest possible N-dimensional shape (hence the name), and has N+1 vertices and N+1 faces, each face being a (N-1)-simplex.)

2A02:C7D:419:2500:8D27:89E7:BBAE:3315 (talk) 06:48, 14 June 2017 (UTC)

It's in the second paragraph, with a link to simplex. Double sharp (talk) 06:58, 14 June 2017 (UTC)