Imagine you have a little robot that holds a pencil. Set it down on a sheet of paper and give it these instructions:

1. Move forward 3 units and turn right.
2. Move forward 1 unit and turn right.
3. Move forward 2 units and turn left.
4. Move forward 1 unit and turn left.
5. Move forward 2 units and turn right.
6. Repeat.

If the robot makes its turns at 90° angles, it will produce this figure:

But, remarkably, if it turns at 120° it will draw this:

Any pair of points define an infinity of ellipses and an infinity of hyperbolas.

The ellipses do not touch one another, nor do the hyperbolas.

But every ellipse meets every hyperbola at a right angle.

In a 1769 letter, Ben Franklin describes a magic square he devised in his youth. The magic total of 260 can be reached by adding the numbers in each row or column, as in a normal magic square. But “bent rows” (shaded) produce the same total, even when “wrapped across” the border of the table. This works in all four directions.

Further: Half of each row or column sums to half of 260, as does any 2×2 subsquare. And the four corners and the four center squares sum to 260. (Alas, the main diagonals don’t, so this doesn’t strictly qualify as a magic square by the modern definition.)

Interestingly, no one knows how Franklin created the square. Many methods have been devised, but none apparently as quick as his, which he claimed could generate them “as fast as he could write.”

Take any Platonic solid, join the centers of its faces, and, charmingly, you get another Platonic solid. The cube and the octahedron produce one another, as do the dodecahedron and the icosahedron, and the tetrahedron produces another tetrahedron.

Bonus factoid: If you inscribe a dodecahedron and an icosahedron in the same sphere, the dodecahedron will occupy more of the sphere’s volume. It has fewer faces than the icosahedron, but its faces are more nearly circular, so it fits the sphere more snugly.

See The Pup Tent Problem.

Let us take a piece of string. In the first half minute we shall form an equilateral triangle with the string; in the next quarter minute we shall employ the string to form a square; in the next eighth minute we shall form a regular pentagon; etc. ad infinitum. At the end of the minute what figure or shape will our piece of string be found to have assumed? Surely it can only be a circle. And yet how intelligible is that process? Each and every one of the polygons in our infinite series contains only a finite number of sides. There is thus a serious conceptual gap separating the circle, as in the limiting case, from each and every polygon in the infinite series.

— Jose Amado Benardete, Infinity: An Essay in Metaphysics, 1964

Turn each of these palindromes “inside out” and their sum remains the same:

13031 + 42024 + 53035 + 57075 + 68086 + 97079 = 31013 + 24042 + 35053 + 75057 + 86068 + 79097

Remarkably, this holds true even if you square or cube them:

13031² + 42024² + 53035² + 57075² + 68086² + 97079² = 31013² + 24042² + 35053² + 75057² + 86068² + 79097²

13031³ + 42024³ + 53035³ + 57075³ + 68086³ + 97079³ = 31013³ + 24042³ + 35053³ + 75057³ + 86068³ + 79097³

From Albert Beiler, Recreations in the Theory of Numbers, 1964.