The Dilettante – More On Infinity

One other fascinating discovery about infinity in One Two Three… Infinity that was new to me is that the number of points on two lines of any length is the same. Also, that the number of points on a plane, and even three dimensional space is the same.

First of all, what I mean by “number of points… is the same” is what you would naturally think: They can be put in a 1:1 (one to one) relationship, for example:

Set 1: A, D, A, M, Z
Set 2: J, U, L, I, E

Set 1 has five letters and set 2 has five letters. We are only concerned with how many items there are in each set, and not what the letters are. The fact that the letter ‘A’ is repeated twice in set 1 and that neither set has any letter in common are unimportant. To see if the two sets are the same size, or if one is bigger than the other, we pair off the items in the two sets in any order we choose:

J - Z
U - M
L - A
I - D
E - A

We find that both sets are the same size because they can be put in a 1:1 relationship.

Simple enough. So, here is the mind blowing visual proof that two lines have the same number of points:


mind blown

In this diagram the two lines of different length AB and AC are joined at A. The line CB connects the endpoints, and every line parallel to CB, such as DE, connects a unique point on AB with a unique point on AC and vice versa. So, even though there are an infinity of points on both lines, they can be put in a 1:1 relationship.

In case that’s a little too informal, let me just add that CB and all its parallels are just graphs of a line function. You don’t even need to know exactly what the function looks like, just that the input is a point on one line and the output is a point on the other, and that for every input there’s a unique output. To me, it seems entirely counter intuitive, but the logic is inescapable, two lines of unequal length have the same number of points.

End of part 1. Coming in Part 2 – Use this one weird trick to map all the points on a plane to all the points on a line.

Review – One Two Three… Infinity by George Gamow






The Dilettante on Living in a Möbius World

Currently reading One Two Three… Infinity by George Gamow.

Here is a little science demonstration inspired by a section on the topology of the cosmos.

Some things I discovered while working on this video:

You can’t make the kind of Möbius strip that I did by nesting two Möbius strips together: you can’t nest two Möbius strips and have one follow the contours of the other all the way around.

If you construct this, and pull it apart so that it hangs loosely as a single loop of paper, it’s very difficult to put it back the way it was. It’s kind of a puzzle.

Here are some other cool Möbius strip related videos I like:

A superconducting vehicle that runs on a magnet covered Möbius strip.

One of my favorites, Vi Hart, showing off a Möbius music box.

This awesome 3D printed round Möbius strip.