• An integer, n, is a Fibonacci number if and only if either 5n^2+4 or 5n^2-4 is a perfect square. 

• There are infinitely many "sizes" of infinities.

Half-explanation: Let me convince you that at least some infinities are bigger than others. Consider the rational numbers, numbers that can be written as fractions (e.g., 0/1, 1/1, -1/1, 2/1, -2/1, 2/2, -2/2, 1/2, -1/2, 3/1, -3/1, 3/2, -3/2, 3/3, -3/3, 2/3, -2/3, 1/3, -1/3 and so on). Clearly there are infinitely many such numbers but they can be listed and will all appear at some point in a list following a pattern such as the one above. For this reason, we say that the rational numbers are "countably" infinite since we can "count" along the list of them. In this way, the rational numbers are the same size as the integers and as the natural (whole) numbers.

Real numbers don't have such a way of listing them in their entirety. A nice property of real numbers is that they are exactly the numbers that can be written down in (not-necessarily finite) decimal form (e.g., 1.00000000001 is a real number, and so is pi ≈ 3.141592654...). 

So suppose there was a way of listing every real number, then consider the number whose first digit past the decimal point is the first digit past the decimal point of the first number in our list plus 1 (or if the digit was 9, make the digit of our considered number 0) and whose second digit is the second digit of the second number on our list plus 1 (with the same caveat) and whose third digit is the third digit of the third number on our list plus 1 (again with the same caveat) and so on for the entire list. Well, this constructed number is written in decimal form and is therefore a real number, but its first digit disagrees with the first digit of the first number on our list so it is not equal to that number on our list, the second digit disagrees with second digit of the second number on our list so it is not equal to that number on our list, and for the same reason it is not equal to the third number on our list, nor the fourth, the fifth, nor the sixth. In fact, this real number that we have constructed is not equal to ANY of the real numbers on our list, and so our list must be incomplete. 

No matter how many times we add these constructed numbers somewhere in our list, there's no hope of completing the list of real numbers because we can just follow the same construction over and over again. Thus there is no way to list out all of the real numbers! For this reason, we say that the real numbers are "uncountably" infinite, and thus larger than the rational, integer, and whole numbers. Turns out that the real numbers end up being the same "size" (this is typically called cardinality) as the irrational numbers, the complex numbers, and (as it turns out) the set of subsets of the rationals (or the set of subsets of the whole numbers). 

To get an infinite set of the "next size up" we can also just take the set of subsets of that infinite set. So, the set of subsets of the real numbers is even bigger than the real numbers, and the set of subsets of the set of subsets of the real numbers is again even larger than the set of subsets of the real numbers. Thus we can see that there are at least countably infinitely many kinds of infinity... The continuum hypothesis states that there are no sizes of infinity between the size of a set and the size of the set of subsets of that set and thus there are exactly countably infinitely many kinds of infinity under the continuum hypothesis.

• For F(n), the n^th Fibonacci number, as n goes to infinity, F(2^{2n}) and F(2^{2n+1}) converge to square roots of -3/5 in the 2-adic numbers. This requires a bit of background to understand and may be the subject of a future coding project, but is very strange even once you understand it.

• Pythagorean time travel: The idea that light always travels at the same speed relative to an observer no matter how fast that observer moves can be used along with the Pythagorean theorem to prove part of special relativity (the time part of the Lorentz transformations) showing that if you were to travel near the speed of light time would seem to quicken around you from your point of view while other observers would see your time seem to slow down (i.e., you would be aging more slowly than those travelling more slowly than you). In this way, you would be able to travel to the distant future, but unfortunately there would be no way to use this same method to travel back to present-day. This gives a new meaning to live fast, die young...

• Any parabola fills exactly 2/3 of the area of any bounding rectangle (a rectangle that intersected by the parabola at two consecutive vertices of the rectangle and at the vertex of the parabola). 

• There are exactly 800 2x2 binary de Bruijn matrices of size 5x5. (Such as my logo) See my doodler for more information and to try to find one yourself.

• By starting with a line segment, then taking out its middle third and leaving two thirds behind then taking out the middle third of those thirds then taking out the middle third of the remaining thirds and so on forever, one arrives at the (middle-thirds) Cantor set. This object is an example of a fractal and has many related fun facts:

  • The middle-thirds Cantor set is self-similar meaning that it contains multiple copies of itself in its entirety. More specifically, when we triple the length of the starting line segment, we get exactly two copies of our originally-sized Cantors sets. This property leads to the next fun fact:

  • The middle-thirds Cantor set has (Hausdorff) dimension ln(2)/ln(3) which is roughly 0.63. 

Explanation: With normal shapes, like a normal line segment (a 1-dimensional object), when we triple the length of the line segment we expect its length to be multiplied by 3 (and rightly so), and when we triple the side length of a square (a 2-dimensional object) we expect its area to be multiplied by 3^2, when we triple the side length of a cube (a 3-dimensional object) we expect its volume to be multiplied by 3^3. So, the power of 3 we multiply the "amount" of  object by when tripling length in a way reports the dimension of the object. As we saw when we triple the length of the middle-thirds Cantor set, we only multiple the amount of Cantor set by 2. What power of 3 is 2? It is precisely the ln(2)/ln(3)-th power of 3.

• The middle-thirds Cantor set has length 0, but has uncountably many points inside of it. Each of these points are completely disconnected from one another yet arbitrarily close to another one of these points, and even further strangely, for that pair of arbitrarily close points, you could draw a two circles each containing one of those points so small such that the circles don't overlap at all.

• A similar fractal to the middle-thirds Cantor set is the four-corners Cantor set in which one starts with a square then cuts away from that square to leave only four squares in the corner each with side length 1/4 of the original, then cuts away and leaves on the quartered-side length corner squares of those squares and so on. This object shares many of the same properties of the above middle-thirds Cantor set, but, interestingly, when I quadruple the side length of the original square and repeat the same process I get four copies of the original size four-corners Cantor set. Following our discussion of dimension from above, this means that the four corners Cantor set has dimension exactly 1.

• Folding a thin, but long piece of paper in half repeatedly in the same direction then unfolding each crease to be a 90 degree angle (in the same direction as the fold) gives rise to another fractal called the Dragon Curve for its loose resemblance to a dragon. The completed fractal (after infinitely many folds) is actually an example of a space filling curve. It can tile the plane, has dimension exactly two, and area exactly equal to half the length of the starting strip of paper.

• If we construct a regular pentagon with hinges at the vertices, then the space of possible positions of that pentagon is in a precise way exactly like the surface of a donut with four holes in it (Where we consider a typical donut to only have the one hole).

• There is no way to comb the hair on a (fully-haired) kiwi (the fruit) such that the kiwi doesn't have a cowlick. 

• No matter how you stir a drink, at least one point in the drink will end up exactly back where it started. (Brouwer's Fixed Point Theorem)

• Even though integrals are typically thought of as undoing derivatives in a way, using complex analysis techniques, it is actually possible to compute a derivative by instead taking an integral.