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:

lines2

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.

The Dilettante – Different Types of Infinity

Different Types of Infinities (Wherein I will try to convince my friend, Jeff, that there are different types of infinities, and that some are ‘bigger’ or ‘more abundant’ than others.)

I’m going to concentrate on showing that integers and real numbers fall into two different categories of infinity. This is an informal proof, meaning that it lacks the rigor that would qualify as a proof for a mathematician, but should (hopefully) be convincing to a layperson. If you are interested in delving deeper, I recommend starting with this video by Vi Hart.

The guy who did the breakthrough work in the mathematics of infinity was Georg Cantor. Before Cantor came along infinity was taken to be a single concept of numbers going on forever, but Cantor showed that the picture was much more complicated and weird. In fact, some of the leading mathematicians of Cantor’s day rejected Cantor’s work and pilloried him.

The Proof

First, integers can be seen as a special case of real numbers that have all zeros after the decimal. We typically ignore the decimal when writing down integers, but another way to write the number 5 is 5.00000000000000… and 3843974937 is 3843974937.00000000000000… (the ellipses “…” denote that the zeros repeat forever). You can see in this diagram that the integers are a subset of the rationals which are in turn a subset of the reals. So, although both sets are infinite, one is a subset of the other.

Now, let’s look at a line segment (i.e. a line of finite length) on the number line, even the smallest line segment we can imagine, will encompass an infinity of real numbers, but only 0 or a finite number of integers.

For example, this line segment from 3.10… to 3.20… contains 0 integers, but an infinity of real numbers, including the irrational number, π (3.1415926…).

Num Line 1

This line segment, from 1.0… to 1,000,000.0… contains 1,000,000 integers, and an infinity of real numbers.

Num Line 2

Cantor developed several proofs, including the diagonal argument, showing the one to one relationship between integers and rationals, and also how the reals cannot be put into a one to one relationship with the integers. There are plenty of websites and videos that explain this concept. So, I’m going to present my own ‘cinematic proof':

Imagine your are in a desert that stretches to the horizon in all directions. Each grain of sand represents a real number, and you are searching for the sand grains that are integers. You are about to pass out from the heat when you see something in the distance. It starts as a dot wavering in the heat waves but eventually you can make out that it’s a person on camelback. Eventually, Omar Sharif is standing before you. He presents you with an easy to follow map to the location of all the integers in the desert, and a scoop that can be adjusted in size to scoop as many of the integers as you want. When you scoop the sand and sift out the integers, you wind up with a small pile, and the sand you discard is a dune that piles up to the sky. That is how the integers and real numbers relate to each other.

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.

The Dilettante: Draw a Hypercube

I learned to draw this hypercube from Rudy Rucker’s Geometry, Relativity and the Fourth Dimension. I highly recommend this book, and wish I had read it as a high school senior or college freshman.

Geometry, Relativity and the Fourth Dimension by Rudy Rucker
Geometry, Relativity and the Fourth Dimension

The hypercube figure is on the cover of the book. According to Rucker, “This design for the hypercube is taken from a little 1913 book, A Primer of Higher Space, by Claude Bragdon, an architect who incorporated this and other 4-D designs into such structures as the Rochester Chamber of Commerce Building.”

At first it looks difficult to draw, but with a little practice, you can actually draw this freehand. Here is how I do it:

First, draw a cube:

Hypercube step 1

That’s pretty easy. I’m using grid paper here, but if you have a steady hand and a good eye, you don’t need it.

Now, draw a second cube with the same dimensions, down and to the right of the first cube. Make them overlap so:

Especially notice the position of the squares that make up the the “front/back” portion of the second cube. Make those squares overlap the the squares from the first cube in the same way. This can be a little tricky at first, but use a different color to draw the second cube, and it will be easier.

Finally, connect the corners of the first cube with the corresponding corners of the second cube:

I’ve used a third color to connect the corners. This can get confusing, but just focus on one pair of corners at a time. The outside corners are the easiest, so start there, and that will give you a way to proceed with the more overlapping parts.

Here’s one I drew in Inkscape with shaded sides:

Hypercube

One fun way to get lost in this figure is to try to count all the cubes that make it up.