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Geometry is one of the oldest branches of mathematics, and many famous problems have been proposed and solved in its long history.

What I would like to know is: What is the oldest open problem in geometry?

Also (soft questions): Why is it so hard? Which existing tools may be helpful to handle it? If twenty great geometers of today gathered to work together in the problem, would they (probably) be able to solve it?

P.S. The problem can be of any area of geometry (discrete, differential, etc...)

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    $\begingroup$ It's impossible to know for sure, so what could happen is that after the first answer, other answers would be about the same problem or about older problems (if one can find)... Sorry for my English. $\endgroup$
    – Sudoku Polo
    Aug 8, 2014 at 4:46
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    $\begingroup$ Quick link: mathoverflow.net/a/27112 $\endgroup$
    – Chris Culter
    Aug 8, 2014 at 5:04
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    $\begingroup$ Squaring the circle but it has been found impossible... $\endgroup$
    – Offirmo
    Aug 8, 2014 at 22:13
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    $\begingroup$ @Offirmo: So it's not an open problem. $\endgroup$
    – TonyK
    Aug 9, 2014 at 11:21

7 Answers 7

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Problem:

Does every triangular billiard have a periodic orbit?

For acute triangles, the question has been answered affirmatively by Fagnano in 1775: one can simply take the length $3$ orbit joining the basepoints of the heights of the triangle.

enter image description here

For (generic) obtuse triangles, the answer is not known in spite of very considerable efforts of many mathematicians. Apparently, A. Katok has offered a $10.000$\$ prize for a solution of this problem.

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    $\begingroup$ +1 for a purely geometric problem. Do you know when the problem was first formulated? $\endgroup$
    – Karolis Juodelė
    Aug 8, 2014 at 12:06
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    $\begingroup$ @KarolisJuodelė In the presented form I don't know, though it is definitely one of the best known problems in the theory of dynamical systems. The original motivation of Fagnano was completely different: he was trying to solve an optimization problem of minimizing the perimeter of a triangle inscribed into another triangle. $\endgroup$
    – Start wearing purple
    Aug 8, 2014 at 12:19
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    $\begingroup$ A recent paper extends the affirmative result to all triangles with largest angle less than $100^\circ$: math.brown.edu/~res/Papers/deg100.pdf $\endgroup$
    – mjqxxxx
    Aug 8, 2014 at 15:52
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    $\begingroup$ @syllogismos Yes. But this is just one example, in principle we could imagine periodic trajectories more complicated than triangles. However if I remember correctly the existence of such more complicated periodic orbits is not proven even for acute triangles. $\endgroup$
    – Start wearing purple
    Aug 9, 2014 at 23:13
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    $\begingroup$ I would like to see an animation for a periodic orbit in an obtuse triangle. $\endgroup$
    – Ziyuan
    Aug 12, 2014 at 16:30
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One of my favourites because it's just so simple to state. Discussed in this MO question and related to an also very interesting, though solved question/puzzle which I posted here a while ago. I'm not sure when the problem was first stated but it could certainly have been understood by even the earliest mathematicians.

Can a disk be tiled by a finite number of congruent pieces (You can rotate or flip pieces onto each other) such that the center of the disk is contained in the interior of one of the pieces?

So far, the only kinds of tilings of the disk by congruent pieces, which we know of, all have the center of the disk lying on the boundary of more than one piece. Here are some examples which fail to have the center in the interior of one of the pieces:

enter image description here From Robert Israel's answer

enter image description here From robjohn's answer

I should add that by 'piece' we mean some nice subset of the disk such as homeomorphic to a disk itself, and equal to the closure of its interior.

By 'tiling' we mean the union of the pieces should be the entire disk, and the intersection of any two pieces should be contained within the union of the boundary of the pieces.

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    $\begingroup$ What does this have to do with the question? Maybe it could have been understood, but if there's no record of it having been asked more than a few years ago it doesn't qualify. $\endgroup$
    – Robert Israel
    Aug 13, 2014 at 1:29
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The $n$-body problem is an ancient problem that was originally a problem of Euclidean geometry (which was originally identified with the study of "space" --- physical space and abstract geometry were not conceptually separated until the modern era). The problem was to determine the motion of $n$ celestial interacting through gravity. It traces its origins to the ancient Greek astronomers, was studied extensively by Kepler, Newton, Poincaré, and continues today. A mathematics competition was posed in 1885 by King Oscar II of Sweden and Norway offered a prize for its solution, which was awarded to Poincaré, although the problem was not solved. Perhaps due to its difficulty the $n$-body problem contributed towards the rise of many different areas of mathematics over the centuries, including calculus (through Newton's work on it), perturbation theory, and chaos.

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    $\begingroup$ How exactly did Greek astronomers think about gravitational interaction of bodies, without knowing the laws of gravity? I don't think it's fair to call a question similar to "how do planets move" equivalent to the n-body problem. $\endgroup$
    – Karolis Juodelė
    Aug 8, 2014 at 12:03
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    $\begingroup$ @Karolis Any question asking for an oldest idea is implicitly (and necessarily) allowing for the answer to account for shifts in terminology and epistemology. Otherwise no answer could date back more than a few dozen years, making the entire exercise boring. $\endgroup$
    – zibadawa timmy
    Aug 8, 2014 at 14:00
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    $\begingroup$ @zibadawatimmy: shifts in terminology aside, is there evidence that the Greeks had any notion of Newtonian dynamics? I don't think it's possible to formulate the $n$-body problem without that. $\endgroup$
    – Will Orrick
    Aug 8, 2014 at 17:54
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    $\begingroup$ @zibadawatimmy: well, in that case the ancient Greeks considered lightning, and I imagine some of them wondered how it works although I don't have the reference, which means providing a theory of Quantum Electrodynamics was (probably) an open problem in vector calculus since antiquity. An absurdity, but you have to draw the line somewhere as to what is studying the same mathematical question vs. what is studying the same physical question using different mathematics. $\endgroup$
    – Steve Jessop
    Aug 8, 2014 at 22:02
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    $\begingroup$ ... that said, I see the Wikipedia page for the n-body problem states that determining what the forces are between them is part of the so-called n-body problem. Which I find odd, because so far as the ancients knew (not having determined the forces), the 2-body problem wasn't inherently any easier. Only once you know that all bodies exert forces on one another do you realise that anything over 2.5 bodies poses a new and specific mathematical challenge ;-) $\endgroup$
    – Steve Jessop
    Aug 8, 2014 at 22:11
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Frank Morgan has referred to the least perimeter way to divide the plane into unit areas as the "oldest open problem in mathematics", dating back to the first millenium BC, when a Roman soldier wrote about the bees on his farm having solved the way to "enclose the greatest amount of space".

One might consider the least perimeter way of enclosing a single area (circle) or volume (sphere), which maybe be proven using symmetry. The problem becomes more difficult when enclosing more than one area -- the two dimensional solution (a "double bubble") was proven in 1991 by Joel Foisy (then an undergraduate), and a three dimensional version by Hutchings, Morgan, Ritore, and Ros in 2000.

To even define the problem of dividing the plane into unit areas (since both area and perimeter will be infinite), one takes a limit of the ratio of perimeter to area inside a ball as the radius of the ball increases. Thomas Hales proved that the solution was the hexagonal honeycomb in 1999, meaning it took a bit over 2,000 years to solve.

The three dimensional version of this problem is called the Kelvin Problem, and the current best conjecture was used in the design for the aquatics center at the Beijing Olympics.

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    $\begingroup$ If this was solved in 1999, then obviously it is disqualified from consideration as the oldest currently open problem in geometry. Nevertheless, I think the problem and it's history is neat enough to merit an honorable mention here. It would be nice if you could provide a more detailed description of the problem in your response itself instead of just linking to an outside source (which could be relegated to comment). $\endgroup$
    – David H
    Aug 8, 2014 at 18:00
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    $\begingroup$ Do I understand correctly that, from it was conjectured that the hexagonal honeycomb was the optimal solution, it took 2000 years to prove it? $\endgroup$
    – kasperd
    Aug 9, 2014 at 18:46
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    $\begingroup$ Yes, though the soldier is somewhat Fermat-ish about it, writing "the geometricians prove that this hexagon inscribed in a circular figure encloses the greatest amount of space." $\endgroup$
    – colcarroll
    Aug 9, 2014 at 20:27
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The square peg problem was posed by Otto Toeplitz over a hundred years ago and is still open; see http://www.ams.org/notices/201404/rnoti-p346.pdf

This involves finding four points on a Jordan curve that form a square.

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I do not know if the following might be considered a problem, but certainly it may be called a question. However, even as a question, I do think it is the most ancient interrogative in geometry or, in any case, its philosophical and theoretical bedrock.

What is a (geometric) space?

After all, geometry was born as an attempt to study (from a mathematical point of view) the (physical) space and different mathematical answers to the above (only apparently) innocent question has led (and will probably lead) to the development of different aspects (kinds, if you prefer) of geometry (euclidean geometry, topology, differential geometry, algebraic geometry and so on and so forth). This is why I consider the determination of the notion of a space the archetipical problem in geometry, its inspiring and ultimate goal, its terrific beauty. And, of course, this is a problem which is as open in geometry as the development of geometry is open in Mathematics.

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The oldest problem we'll never really solve:

what is the geometry of physical space ?

Sure, we'll find better approximations to this question, but, at a basic level, I doubt this is ever resolved.

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    $\begingroup$ @mistermarko general relativity is not general. We have quantum mechanics and they are at odds. String theory tries to fix this, but, at the base of it all the compactification problem confounds us. Which of the myriad of Calaubi-Yau's is the right choice, or, is it $G_2$, I lose track, I'm sure they've moved on. It is an ongoing problem which may not have a solution or, even a class of solutions... it's "open" $\endgroup$
    – James S. Cook
    Aug 8, 2014 at 6:25
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    $\begingroup$ @mistermarko In addition to what James said, even if we restrict our attention to situations where the field equations of general relativity are approximately an exact an description of physical space, we still run up against the practical problem of solving the equations. We know solutions to a few special cases, and we have numerical approximation algorithms for a wider range of cases, but there's a much larger set of cases that we don't know much about at all. $\endgroup$
    – David H
    Aug 8, 2014 at 6:36
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    $\begingroup$ Isn't it a problem in physics not geometry? $\endgroup$
    – Ruslan
    Aug 8, 2014 at 7:49
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    $\begingroup$ @ David H About your last statement, surely we can numerical approximate any gravitational situation? $\endgroup$
    – user117644
    Aug 8, 2014 at 7:57
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    $\begingroup$ @Ruslan: Interesting point. Possibly the answer to your question changed over time. If geometry is the study of properties of physical space then it's a problem in geometry to establish them. If geometry is an abstract theory that co-incidentally might have something to do with physical space, then it's a problem in physics. The trick is to identify the historical point at which it changed, when geometers "lost interest" in reality ;-) $\endgroup$
    – Steve Jessop
    Aug 8, 2014 at 9:14

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