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

One Two Three… Infinity by George Gamow was originally published in 1947 and revised in 1961. There have been many advancements in both science and mathematics since, however it remains an engaging introduction to some of the fundamental and fascinating topics in science and mathematics. Being dated is not always such a bad thing: there is a certain excitement about subjects like the existence of other galaxies beyond the Milky Way, that is just taken for granted in popular science writing today. Also, some facts and explanations (e.g. Einstein’s theories and imaginary numbers) are as true today as they were back then.

One glaring omission I noticed was DNA. I don’t know what was revised between the original 1947 version of One Two Three… Infinity and the 1961 edition, but the DNA molecule which had been discovered in the early 1950s, and for which Watson and Crick received a Nobel Prize in 1962, seems like a pretty big oversight. That being said, Gamow speculates on the mechanism and aspects of the molecule responsible for inheritance and the makeup of genes that gives interesting insight into what was known about the subject before the discovery of DNA.

The mathematics chapters had some different ways of explaining some ideas that I had not seen before. One of the satisfying things about reading multiple books by different authors on the same subjects is that you can find explanations that complement or better explain a concept. I’ve seen Cantor’s proof that the Rational Numbers are countable illustrated with a table with numerators on the horizontal and denominators on the vertical axis. Gamow, instead, says: imagine all the fractions with the numerator and denominator that add to 2 (1/1), 3 (1/2, 2/1), 4 (1/3, 2/2, 3/1), etc..

I really enjoyed the chapter on nuclear physics, and it even inspired me to build my own cloud chamber for detecting cosmic rays. There’s also a great section about neutrinos and how they came to be discovered. Another subject that Gamow covers, that I’m always personally fascinated by and that inspired another science demonstration is his chapter on the topology of the Universe.

Every chapter has a gem or two, and Gamow’s style is informal with jokes and humorous illustrations sprinkled about. Although there are contemporary books that may delve more or less deeply into any of the covered topics, One Two Three… Infinity is a classic that (mostly) stands the test of time.