As mentioned in lecture, Hilbert’s third problem was an attempt to understand whether the Bolyai-Gerwien theorem could generalize to

3. THE EQUALITY OF THE VOLUMES OF TWO TETRAHEDRA OF EQUAL BASES AND EQUAL ALTITUDES.

In two letters to Gerling, Gauss expresses his regret that certain theorems of solid geometry depend upon the method of exhaustion, i.e., in modern phraseology, upon the axiom of continuity (or upon the axiom of Archimedes). Gauss mentions in particular the theorem of Euclid, that triangular pyramids of equal altitudes are to each other as their bases. Now the analogous problem in the plane has been solved. Gerling also succeeded in proving the equality of volume of symmetrical polyhedra by dividing them into congruent parts. Nevertheless, it seems to me probable that a general proof of this kind for the theorem of Euclid just mentioned is impossible, and it should be our task to give a rigorous proof of its impossibility. This would be obtained, as soon as we succeeded in specifying two tetrahedra of equal bases and equal altitudes which can in no way be split up into congruent tetrahedra, and which cannot be combined with congruent tetrahedra to form two polyhedra which themselves could be split up into congruent tetrahedra.

Hilbert’s student Max Dehn solved the problem in 1901 with the introduction of what we now call Dehn invariants:

Theorem. If two polyhedra in are equidecomposable into polyhedra, then they have the same volume and the same Dehn invariants.

In 1965, J.-P. Sydler proved the converse of Dehn’s result:

Theorem. Two polyhedra in with the same volume and the same Dehn invariants are equidecomposable into polyhedra.

A couple of years ago, Richard Schwartz, from Brown university, wrote a couple of very nice notes explaining both Dehn’s and Sydler’s theorems. He also developed a Java applet illustrating Sydler’s argument (for his “Fundamental lemma”). They can be downloaded here.

(The nicest presentation of the Bolyai-Gerwein result that I’ve found is in Howard Eves’ “A Survey of geometry“. The text of Hilbert’s original lecture delivered before the International Congress of Mathematicians at Paris in 1900 was expanded to a paper, “Mathematical problems”, Bull. Amer. Math. Soc. 8 (1902), 437–479. It has been recently (I’m old) reprinted, in Bull. Amer. Math. Soc. (N.S.) 37 (2000), no. 4, 407–436 and can be downloaded here.)

43.614000-116.202000

Like this:

LikeLoading...

Related

This entry was posted on Monday, January 23rd, 2012 at 1:32 pm and is filed under 515: Analysis II. You can follow any responses to this entry through the RSS 2.0 feed.
You can leave a response, or trackback from your own site.

(1) Patrick Dehornoy gave a nice talk at the Séminaire Bourbaki explaining Hugh Woodin's approach. It omits many technical details, so you may want to look at it before looking again at the Notices papers. I think looking at those slides and then at the Notices articles gives a reasonable picture of what the approach is and what kind of problems remain […]

It is not possible to provide an explicit expression for a non-linear solution. The reason is that (it is a folklore result that) an additive $f:{\mathbb R}\to{\mathbb R}$ is linear iff it is measurable. (This result can be found in a variety of places, it is a standard exercise in measure theory books. As of this writing, there is a short proof here (Intern […]

I learned of this problem through Su Gao, who heard of it years ago while a post-doc at Caltech. David Gale introduced this game in the 70s, I believe. I am only aware of two references in print: Richard K. Guy. Unsolved problems in combinatorial games. In Games of No Chance, (R. J. Nowakowski ed.) MSRI Publications 29, Cambridge University Press, 1996, pp. […]

Let $C$ be the standard Cantor middle-third set. As a consequence of the Baire category theorem, there are numbers $r$ such that $C+r$ consists solely of irrational numbers, see here. What would be an explicit example of a number $r$ with this property? Short of an explicit example, are there any references addressing this question? A natural approach would […]

Suppose $M$ is an inner model (of $\mathsf{ZF}$) with the same reals as $V$, and let $A\subseteq \mathbb R$ be a set of reals in $M$. Suppose further that $A$ is determined in $M$. Under these assumptions, $A$ is also determined in $V$. The point is that since winning strategies are coded by reals, and any possible run of the game for $A$ is coded by a real, […]

The two concepts are different. For example, $\omega$, the first infinite ordinal, is the standard example of an inductive set according to the first definition, but is not inductive in the second sense. In fact, no set can be inductive in both senses (any such putative set would contain all ordinals). In the context of set theory, the usual use of the term […]

I will show that for any positive integers $n,\ell,k$ there is an $M$ so large that for all positive integers $i$, if $i/M\le \ell$, then the difference $$ \left(\frac iM\right)^n-\left(\frac{i-1}M\right)^n $$ is less than $1/k$. Let's prove this first, and then argue that the result follows from it. Note that $$ (i+1)^n-i^n=\sum_{k=0}^{n-1}\binom nk i^ […]

I think it is cleaner to argue without induction. If $n$ is a positive integer and $n\ge 8$, then $7n$ is both less than $n^2$ and a multiple of $n$, so at most $n^2-n$ and therefore $7n+1$ is at most $n^2-n+1

Let PRA be the theory of Primitive recursive arithmetic. This is a subtheory of PA, and it suffices to prove the incompleteness theorem. It is perhaps not the easiest theory to work with, but the point is that a proof of incompleteness can be carried out in a significantly weaker system than the theories to which incompleteness actually applies. It is someti […]