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3.6 — Geometric Gems: Results Worth Knowing for Their Own Sake
Some geometry is useful and some is simply astonishing. This chapter collects the second kind — theorems that make people sit up, along with the shapes that turned out not to obey the rules everyone assumed. None of it is required for later chapters. All of it is worth having in your head, because these are the results that make good conversation and that show what mathematics feels like from the inside.
1. Monge's theorem
Draw three circles of different sizes anywhere on a page, no one inside another. For each pair of circles, draw the two lines that are tangent to both and cross between them — the "external" tangent lines that touch both circles on the same side. Each pair of circles gives one crossing point where its two external tangents meet.
Three pairs, three points.
Those three points always lie on a single straight line. Always, for any three circles, whatever their sizes or positions.
There is no reason from the two-dimensional picture that this should happen, and trying to prove it in the plane is unpleasant. The proof, due to Gaspard Monge around 1800, is a magic trick:
Lift the picture into three dimensions. Replace each circle with a sphere of the same radius, sitting so the circle is its equator. Now instead of tangent lines, think of a cone wrapped around each pair of spheres, touching both — the apex of that cone sits exactly at the point where the two tangent lines met.
Here is the move. There is a plane that rests on top of all three spheres, like a sheet of glass laid on three balls of different sizes. There is also the original flat page underneath them. Two planes, and two distinct planes meet in a straight line.
Now: each cone's apex lies on both planes. It lies on the page because that is where we drew it, and it lies on the glass sheet because a cone tangent to two spheres has its apex on any plane tangent to both. Three points, all on both planes, and the intersection of two planes is a line. Therefore the three points are collinear.
The two-dimensional statement was hard; the three-dimensional statement was almost trivial. This is one of the most valuable habits in mathematics — a problem that resists in its own setting sometimes dissolves in a larger one. Chapter 3.1 saw the same with the impossible constructions, solved by moving from geometry to algebra.
2. The Pizza theorem, and other slicing surprises
The pizza theorem. Pick any point inside a circular pizza — not necessarily the centre. Cut four straight cuts through that point, at 45° to each other, giving eight slices. Take alternate slices. Whoever takes the alternate set gets exactly half the pizza, regardless of where the point was.
It works for any number of cuts that is a multiple of four (eight slices, twelve slices, and so on) and fails for other counts. It is a genuine theorem with a real proof, and it is the most useful piece of mathematics ever applied to sharing food.
The napkin ring problem is stranger. Take a sphere and drill a cylindrical hole straight through its centre, then measure the height of the remaining ring — the band of material left.
The volume of what remains depends only on that height, not on the size of the original sphere.
Drill a hole through a marble leaving a 6 cm tall ring. Drill a hole through the Earth leaving a 6 cm tall ring. The two rings have identical volume. With the Earth you removed almost everything, and with the marble you removed almost nothing, and the leftovers match exactly.
The volume is \frac{\pi h^3}{6} where h is the ring's height — no radius anywhere in the formula. The reason is that a bigger sphere needs a wider hole to leave the same ring height, and the extra width removes precisely the extra material. Chapter 5.5 lets you compute it in three lines.
3. The Banach–Tarski paradox
This one sounds like a joke and is a theorem.
A solid ball can be cut into five pieces which, moved only by rotation and translation with no stretching, reassemble into two solid balls each the same size as the original.
Not two smaller balls. Two identical copies.
Nothing is wrong with the proof. What is wrong is our assumption that every set of points has a well-defined volume. The five "pieces" are not shapes in any physical sense — they are infinitely scattered clouds of points, so intricate that no consistent volume can be assigned to them at all. When you reassemble them, you are not moving stuff around; you are exploiting the fact that "how much is here" was never defined for these objects.
The construction requires the axiom of choice, a set-theory assumption that lets you pick one element from each of infinitely many sets without specifying a rule for the picking. Most mathematicians accept it because too much useful mathematics depends on it, and this paradox is the price.
What it actually teaches: infinity does not behave like a big number, and intuitions trained on physical objects fail completely there. Chapter 1.7 showed the mild version, where a set can be matched with a proper part of itself. This is the extreme version. And it says something honest about mathematics — that "volume" is a definition we impose, not a property that objects simply have.
4. Fractals: shapes with fractional dimension
A line is one-dimensional, a square two, a cube three. That seems exhaustive. It is not.
Take a line segment and remove the middle third. Take each remaining piece and remove its middle third. Repeat forever. What remains is the Cantor set.
The Cantor set has zero total length (the removed pieces sum to the whole segment) and yet contains uncountably many points — the same infinity as the entire real line, by Chapter 1.7's argument. It is a set of measure zero with the cardinality of the continuum.
Its dimension is \frac{\ln 2}{\ln 3} \approx 0.631. Here is what that means, using a definition that agrees with the ordinary one and extends past it. Scale a line by 3 and you need 3 copies to rebuild it; scale a square by 3 and you need 9 = 3^2; scale a cube by 3 and you need 27 = 3^3. The exponent is the dimension. For the Cantor set, scaling by 3 gives something needing only 2 copies, so the dimension d satisfies 3^d = 2, giving d = \log_3 2 — the logarithm from Chapter 1.4, doing something nobody expected it to do.
Where this stops being a curiosity. In 1967 Benoit Mandelbrot asked how long Britain's coastline is, and found the question has no answer: the smaller your measuring stick, the longer the coast, without limit, because every zoom reveals more inlets. Coastlines, clouds, mountains, lungs, blood vessels, river networks and lightning are all fractal — roughly self-similar across scales — and the fractal dimension is a real measurable number that characterises how rough something is.
Your lungs pack about 70\ \text{m}^2 of surface area into a few litres of volume by branching fractally. That is not an analogy; it is the design.
5. Topology: geometry that does not care about size
Suppose you are allowed to stretch, bend and squash a shape as much as you like, but never tear it or glue it. Which properties survive?
Not length, not angle, not area. What survives is connectivity — how many pieces, how many holes, what is inside what. This is topology, and it is a genuinely different way of looking at shape.
The standard joke is that a topologist cannot tell a coffee mug from a doughnut, and it is exactly right: both are solid objects with a single hole, and one can be continuously deformed into the other.
Euler's formula for polyhedra is the classic result. For any convex solid with flat faces:
V - E + F = 2
Vertices minus edges plus faces. A cube: 8 - 12 + 6 = 2. A tetrahedron: 4 - 6 + 4 = 2. An icosahedron: 12 - 30 + 20 = 2. Every one of them, and any solid you build.
The number 2 is the Euler characteristic, and it is a topological property — it does not change when you deform the solid. Put one hole through the object and it becomes 0. Two holes, -2. The formula counts holes, which is why it survives being stretched.
This has a direct consequence you have seen. A football cannot be tiled with hexagons alone — the count does not work out — and Euler's formula forces exactly twelve pentagons onto any such surface, however many hexagons you add. The classic ball has 12 pentagons and 20 hexagons. Carbon molecules called fullerenes obey the same constraint, always with exactly twelve pentagonal faces.
The four colour theorem. Any map drawn on a flat surface can be coloured with four colours so no two neighbouring regions share a colour. Conjectured in 1852, and proved in 1976 by Appel and Haken — using a computer to check 1,936 cases, which caused a genuine crisis about what counts as a proof if no human can read it. Modern verified proofs have been machine-checked end to end, which some find reassuring and others find sidesteps the objection.
The Möbius strip. Take a paper strip, half-twist it, and join the ends. It now has one side and one edge — run a pen along the surface and you return to the start having covered what looks like both sides without crossing an edge. Cut it down the middle and you do not get two strips; you get one longer strip with a full twist. Cut that down the middle and you get two linked rings.
Conveyor belts and printer ribbons have been built as Möbius strips so that wear is spread over the whole surface rather than one side.
6. Two more results that sound impossible
The hairy ball theorem. You cannot comb a hairy sphere flat without leaving at least one cowlick — a point where the hair stands up or the parting meets. Formally, a continuous non-vanishing tangent vector field on a sphere does not exist.
The immediate consequence: at any moment there is at least one point on Earth where the horizontal wind speed is exactly zero. Not because of meteorology, but because of topology. There is always an eye somewhere.
Interestingly, it is false for a doughnut — you can comb a torus perfectly flat. The obstruction is the sphere's shape, and the same Euler characteristic of Section 5 is what decides it.
The Borsuk–Ulam theorem. At any moment, there exist two points exactly opposite each other on the Earth's surface with identical temperature and identical barometric pressure.
The proof idea is a continuity argument. Consider the function "temperature here minus temperature at the antipodal point". If it is positive somewhere, it is negative at that point's opposite, since the roles swap. A continuous function that is both positive and negative must be zero somewhere in between — the Intermediate Value Theorem of Chapter 5.1. Getting two quantities to match simultaneously needs the full theorem, but the flavour is the same.
The ham sandwich theorem is the same circle of ideas: given any three solid objects in space, there is a single flat plane that cuts all three exactly in half simultaneously. Two slices of bread and a slice of ham, however badly arranged, can always be halved with one cut.
7. Why any of this matters
None of these results will help you at work. Here is why they are in the book anyway.
They show what proof is for. Everything in this chapter is either counterintuitive or plainly unbelievable, and yet each is settled beyond argument. Measurement could never have established any of them. This is the payoff of the machinery Euclid started in Chapter 3.1.
They show that the useful and the beautiful are the same subject. Fractals were a curiosity until they described coastlines, lungs and network traffic. Topology was abstract nonsense until it explained the twelve pentagons on a football, the shape of DNA knots, and the phases of matter that won the 2016 Nobel Prize in Physics. Monge's dimension-raising trick is the same move that makes a hard machine-learning problem easy by adding features.
They are excellent to talk about. Two identical napkin rings from a marble and from the Earth, a wind-free point that must always exist somewhere, a coastline with no definite length — these are the things that make someone who thought they hated mathematics ask a second question.
Part 3 is finished. We have shape, distance, angle and the beginnings of curvature. What we do not have is a way to handle many quantities at once, systematically, in a form a machine can execute. That is linear algebra, and it is the mathematics that quietly runs the modern world.
More places these turn up
The law of cosines is what your phone's GPS solves, several times a second, to turn distances-to-satellites into a position. The addition formulas are how any graphics engine rotates an object, and the single-wave form of §6 is how your headphones' noise cancelling adds an opposing wave to the one arriving at your ear. The catenary is holding up every overhead power line you drive past, and \tanh is inside the model that suggested what you should watch tonight. Radians are not a mathematician's preference but the only unit in which \frac{d}{dx}\sin x = \cos x, which is why every piece of scientific software works in them and every one of them is wrong if you feed it degrees.
Next: 3.P — Worked Problems uses these formulas on triangles, circles, waves and hanging cables, step by step.