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Hometetrahedron

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# tetrahedron

# 1 Definition

A *tetrahedron* is a polyhedron with four faces, which are
triangles. A tetrahedron is called non-degenerate if the four
vertices do not lie in the same plane. For the remainder of this
entry, we shall assume that all tetrahedra are non-degenerate.

If all six edges of a tetrahedron are equal, it is called a
*regular tetrahedron*. The faces of a regular tetrahedron are
equilateral triangles.

# 2 Basic properties

A tetrahedron has four vertices and six edges. These six edges can be arranged in three pairs such that the edges of a pair do not intersect. A tetrahedron is always convex.

In many ways, the geometry of a tetrahedron is the three-dimensional analogue of the geometry of the triangle in two dimensions. In particular, the special points associated to a triangle have their three-dimensional analogues.

Just as a triangle always can be inscribed in a unique circle, so too a tetrahedron can be inscribed in a unique sphere. To find the centre of this sphere, we may construct the perpendicular bisectors of the edges of the tetrahedron. These six planes will meet in the centre of the sphere which passes through the vertices of the tetrahedron.

The six planes which connect an edge with the midpoint of the opposite edge (see what was said about edges coming in pairs above) meet in the barycentre (a.k.a. centroid, centre of mass, centre of gravity) of the tetrahedron.

# 3 Mensuration

Formulas for volumes, areas and lengths associated to a terahedron are best obtained and expressed using the method of determinants. If the vertices of the tetrahedron are located at the points $(a_{x},a_{y},a_{z})$, $(b_{x},b_{y},b_{z})$, $(c_{x},c_{y},c_{z})$, and $(d_{x},d_{y},d_{z})$, then the volume of the tetrahedron is given by the following determinant:

$\pm\frac{1}{6}\left|\begin{matrix}a_{x}&a_{y}&a_{z}&1\\ b_{x}&b_{y}&b_{z}&1\\ c_{x}&c_{y}&c_{z}&1\\ d_{x}&d_{y}&d_{z}&1\\ \end{matrix}\right|.$ |

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51E99*no label found*

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## Comments

## n-agons and polyhedrons

The definition of n-agons and polyhedrons can be exposed in only one entry. I don't see the utility to display so many entries for definitions that are analogous and, beside the point, trivial.

## Re: n-agons and polyhedrons

the proper way, of course

is to mention this on the "polyhedron" entry

nd then addind "tetrahedron" to the "also defines" field on the edit entry dialog

f

G -----> H G

p \ /_ ----- ~ f(G)

\ / f ker f

G/ker f

## not necessarily

As you can see, there are a lot of things which can be said about tetrahedra which are not special cases of general facts. Therefore, it is good for the tertrahedra to have their own special place.

## I just read your article and

I just read your article and it was totally great, it contains a lot of useful ideas, it is also written in organize manner,thanks for sharing this kind of article.

## I like it

I think I have a greater understanding of a tetrahedron now, your explanation was simple enough. I was thinking of studying mathematics at university, but instead opted to go into computing, repair diagnostics, sys backup & recovery etc... which has turned out to be most helpful for me, at least.

## pseudoprimes in k(i) (contd)- a small by-product

A small by-product of research in area of pseudoprimes

^{}in k(i): Take a product^{}of two numbers each with shape 4m+3. Let x be this composite number. x is pseudo to base (x-1).Examples 21, 33, 57 etc. (20^20-1)/21 yields a rational integer.## Fermat's theorem in terms of matrices.

Let X be a square matrix in which each element is an odd prime. Then (a^(X-I)-I)/X yields a square matrix in which the elements belong to Z. Here a is co-prime with each element of X. Also I is the identity matrix.

## A request to Dr. Puzio

I use pari software and sometimes I would like to display the calculations/programs on the space for messages; however, I am unable to paste them. Would be glad if this and adding files are enabled.

## A puzzle

Fermat’s theorem works in terms of square matrices; however Euler’s generalisation of Fermat’s theorem in terms of matrices does not seem to be true.