My question is a direct continuation of my already posted question https://hsm.stackexchange.com/questions/6772/did-gausss-expression-for-the-differential-of-the-hyperbolic-volume-of-the-tetr (asked on "history of science and mathematics" stackexchange). I simply didn't find any sources that say that Gauss's result was a nonesense; in his commentary on Gauss's relevant note, Stackel doesn't say Gauss was mistaken (apart from a factor of $1/2$ that was missing fron his expression for the volume differential), and in particular, the book "Mathematics and Its History",p.379, emphasizes that Gauss "did have many of the results of non-euclidean geometry by this time, including the answer to the volume problem he raised to test his young rival (see Gauss (1832 - Cubierung der Tetraeder)". I ask again a very similar question simply because i didn't get an answer to the title question in my previous post, and since questions that have answers are likely to "go the graveyard", i preffered to ask it again.

I really want to place Gauss's result in historical context, but the conceptual framework of hyperbolic geometry is as new to me as it was to Gauss's contemporaries, so i can't succeed in this task without help. My previous post helped me understand the Gauss's formula for the orthoscheme tetrahedron; it connects the length of the side 24 with the angle 341 by the formula:

$$\alpha^2\cdot cotg^2341 - \beta^2\cdot tanh^2(l_{24}) = 1$$

when: $$\alpha = cotg431,\quad \beta = cotg 234.$$

Literally speaking, Gauss's formula says the length of the side 24 is an inverse hyperbolic function of a trigonometric function of the angle 341. Now, i understand the method of exhaustion is universal and doesn't depend on type of geometry; whether the geometry is hyperbolic, elliptic or euclidean, one can find volumes by dividing it into slices and then integrate them. But i don't know how to move from the Gauss's expreesion for the differential:

$$\partial \Delta = - \frac{{1}}{{2}} l_{24}\cdot \partial A_{341},$$

to the volume function; in particular, i lack an expression for the area of the face 341 as a function of the length 24. If anybody can help me with that, then i believe i'll be able to derive an expression for the volume function.

Perhaps I'm not appreciating well the difficulty of the subject (i know three-dimensional hyperbolic geometry is a pretty advanced topic) and there are very few people who can answer my question, but i won't give up until i'll exhaust all of my options.

  • Bolay? do you mean Bolyai? – Gerry Myerson Dec 14 '17 at 11:43
  • 1
    I corrected the spelling of his name. – user2554 Dec 14 '17 at 14:53
up vote 3 down vote accepted

I am not sure of the notation, but I assume this can be derived from the Schlafli formula for the volume of a tetrahedron (so this seems to indicate that Gauss knew Schlafli's formula three quarters of a century prior to Schlafli):

$$ dV = -\frac12 \sum_{ij}l_{ij} d \alpha_{ij},$$ where $l$ is the length of the edge, and $\alpha$ is the dihedral angle of the edge. Notice, in particular, that if the dihedral angle at an edge does not change, then that edge does not contribute to the sum. Further note that if you look at the link of the vertex $1$ (wlog), this is a spherical triangle, whose angles are the dihedral angles $12, 13, 14$ while its sides are the face angles of the three adjacent faces. By the Gauss-Bonnet theorem (note the first author), the variation of the area of the face is equal to minus the sum of the variations of the angles.

Put this all together, and you should get Gauss' formula. As for Schlafli's formula, there are many nice proofs, a simple geometric one by Vinberg (which appeared in the Geometry of Spaces of Constant Curvature survey), and a very pretty analytic one by Hellmuth Kneser, which appeared in Deutsche Mathematik, and thus is hard to find, but there is a more recent exposition a paper of Feng Luo https://arxiv.org/abs/math/0412208.

  • I voted your answer because it really helped me understand things about Gauss's formula (now i know it's form is similar to the one given by Schlafli, and that helps me a lot). Thanks!, but i still need further explanation. In particular, as far as i know, Gauss meant hyperbolic geometry when he talked about "non-euclidean geometry", but when you mentioned spherical triangle it made suspect that perhaps we are not talking about the same thing. – user2554 Dec 15 '17 at 7:55
  • In addition, since i don't understand well the rimannnian theory of 3-manifolds, i have a basic misunderstanding: in three-dimensional hyperbolic spaces with constant space curvature k, the Gauss-Bonnet theorem can be applied to figures with volume, but does it mean it can be applied to figures with finite area but zero volume? i ask because by analogy with the Gauss-Bonnet theorem for 2-manifolds, it cannot by applied to cross sections (sections of one dimension lower). – user2554 Dec 15 '17 at 8:04
  • @user2554 The link of a vertex of a polyhedron in any geometry is a spherical triangle. As for Gauss-Bonnet, in its original form (due to Gauss) it is stated for triangles. In any case, area as angle defect is a fundamental fact of hyperbolic geometry. – Igor Rivin Dec 15 '17 at 16:51
  • Sorry but i'm not a professional mathematician - i asked my question here simply because its the only place where there are experts to answer my question. As far as i know, "link" is a concept from knot theory, and i dont know the concept of a "link of a vertex of a polyhedron". What is it? – user2554 Dec 15 '17 at 17:31

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