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In school we have learned about objects in $2$-space and in $3$-space, with heavy emphasizes on the properties in $2$-space. My question can be formulated as follows:

What would we have learned in school, if the geomtry lessons didn't essentially restrict to the geometry of simplices in the euclidean plane, but fully generalized these inspections for $d$-dimensional simplices?

Whereas knowledge on the geometry of a triangle is taken for granted, I admit I don't know about corresponding results for a tetrahedron, (e.g.: What is the sum of the angles between faces?) not to speak of the higher dimensional analogues.

Whereas this is elementary, I don't think this is completely trivial. Do you know articles or other resources which develop the geometric theory of $d$-simplices in the elementary way I described?

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I guess the thing is that three dimensions are already vastly more complicated than two dimensions. One instance of this is Hilbert's third problem which you might find interesting. –  t.b. Aug 15 '11 at 11:27
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My note "Heron-Like Results for Tetrahedral Volume" (daylateanddollarshort.com/math/pdfs/heron4tet.pdf)discusses some aspects of tetrahedra --such as Laws (plural!) of Cosines for faces and dihedral angles-- that echo properties of triangles. Higher-dimensional analogues can get messy, but at least the Pythagorean Theorem is nice everywhere: for a "right-corner" $d$-simplex, the square of the content of the hypotenuse-cell is equal to the sum of the squares of the contents of the leg-cells. ($H^2 = L_1^2 + L_2^2 + \cdots + L_d^2$) –  Blue Oct 14 '11 at 16:17
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For those who can read German there is a very recent book about tetrahedra: Elementare Tetraedergeometrie - Eine Einführung in die Raumgeometrie by Heinz Schumann.

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For a start, you could try the Wikipedia article on tetrahedra.

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