Experiments you can run yourself.
Every surface without "holes" can be continuously deformed into a sphere.
A Möbius strip cut along its centre line gives one longer band, not two.
The golden ratio equals about 1.618 (the golden number φ).
Cutting a cone with a plane parallel to its slant line gives a parabola.
Mathematically, one ball can be reassembled into two identical ones — assuming the axiom of choice.
The area of the square on the hypotenuse equals the sum of the areas of the squares on the legs.
For any convex polyhedron, V − E + F always equals 2.
The successive hypotenuses in the spiral of Theodorus have lengths equal to the successive square roots √n.
Parallel lines crossing the arms of an angle cut them into proportional segments.
Only a dilation changes a figure's size; isometries preserve distances.
You cannot construct, with compass and straightedge, a square whose area equals a given circle.
Trisecting an arbitrary angle with compass and straightedge is impossible (proven).
The motion of three gravitating bodies cannot be predicted by a formula forever — it can be chaotic.
The sum of the angles of a triangle drawn on a sphere is greater than 180°.
The Klein bottle has no separate inside and outside — it is one-sided.
On the whole Earth a horizontal wind cannot blow everywhere at once — there is always a windless spot.
When you stir coffee in a cup, some point of the liquid always returns exactly to its place.
The area of a polygon with vertices on a grid is found by counting dots: P = i + b/2 − 1.
Not only the circle has constant width — the Reuleaux triangle can roll too.
The angle trisectors of any triangle always form an equilateral triangle.
A single flat cut can simultaneously halve bread, cheese and ham — arranged any way at all.