elegantly constructed proofs through visual inspection
Visual proof for Euler’s Identity:
Starting at e0 = 1, travelling at the velocity i relative to one’s position for the length of time π, and adding 1, one arrives at 0.
Graphical representation of Collatz conjecture depicting all numbers with an orbit length of 20 or less
The Collatz conjecture is an unsolved conjecture in mathematics named after Lothar Collatz, who first proposed it in 1937.
Take any natural number n. If n is even, divide it by 2 to get n / 2, if n is odd multiply it by 3 and add 1 to obtain 3n + 1. Repeat the process indefinitely. The conjecture is that no matter what number you start with, you will always eventually reach 1. It has been called “Half Or Triple Plus One”, sometimes called HOTPO.
Penrose tiles in ancient Islamic art (full article)
A Penrose tiling is a nonperiodic tiling generated by an aperiodic set of prototiles named after Sir Roger Penrose, who investigated these sets in the 1970s. Because all tilings obtained with the Penrose tiles are non-periodic, Penrose tiles are considered aperiodic tiles. A Penrose tiling may be constructed so as to exhibit both reflection symmetry and fivefold rotational symmetry, as in the diagram at the right.
A Penrose tiling has many remarkable properties, most notably:
- It is nonperiodic, which means that it lacks any translational symmetry. More informally, a shifted copy will never match the original exactly.
- Any finite region in a tiling appears infinitely many times in that tiling and, in fact, in any other tiling. This property would be trivially true of a tiling with translational symmetry but is non-trivial when applied to the non-periodic Penrose tilings.
- It is a quasicrystal: implemented as a physical structure a Penrose tiling will produce Bragg diffraction; the diffractogram reveals both the underlying fivefold symmetry and the long range order. This order reflects the fact that the tilings are organized, not through translational symmetry, but rather through a process sometimes called “deflation” or “inflation.”
Various methods to construct Penrose tilings have been discovered, including matching rules, substitutions, cut and project schemes and coverings.