Monarch Butterfly captured in slow motion on a bougainvillea outside my house. Spring in Southern California.
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.
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…).
This line segment, from 1.0… to 1,000,000.0… contains 1,000,000 integers, and an infinity of real numbers.
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.
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.
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:
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:
One fun way to get lost in this figure is to try to count all the cubes that make it up.
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:
Some musings on slavery and colonialism from The Cartoon History of the Modern World, Part II.
In the late 1700’s, the British empire circled the globe. A couple of hold-outs were China; which would trade goods for silver, but never buy anything; and Japan, which after uniting under a single Shogun in 1600, refused all contact with the Western World. Of course, capital being hard to say ‘no’ to, both countries left single ports open to trade.
Although other European countries participated in the slave trade, the British were the most successful. The slave trade was run in a triangle: African slaves were shipped to sugar plantations in the New World, sugar was shipped to New England to be distilled into rum, the rum was then sent back to Africa to trade for more slaves. While that was going on, the British East India Company was busy mismanaging India, and eventually ran the Indian economy into the ground leading to widespread poverty.
There had always been people in Britain that opposed the slave trade, but they were mostly seen as a wacky fringe group. Then, the Zong massacre renewed the anti-slavery activists’ fight, and an anti-slavery MP, William Wilberforce, took up the cause, introducing an anti-slavery bill to Parliament – it was defeated. Every year Wilberforce introduced the bill, and every year it was voted down.
After the British defeated Napoleon, Wilberforce, hit on the idea of outlawing the slave trade on the high seas as an extension of British naval power. By voting to ban the slave trade for all nations, British ships could to stop and board any ship sailing under any nation’s flag. After 20 years, Wilberforce finally got his anti-slavery bill passed. The British didn’t do a lot to stop the slave trade, but over the yeas they managed to free tens of thousands of slaves. The U.S., seeing the reality of the situation, decided to follow the British by voting to ban the slave trade on the open seas in 1808. The American’s did nothing to enforce the ban, however.
After crashing India’s economy, the British hit on the thing that they could sell to China: opium from Indian poppies. This trade was so lucrative that even after the British voted in 1833 to ban slavery throughout the empire, they were able to make up for the loss of revenue with the trade. The opium trade divided China and lead to the Opium Wars which took 60 million lives.
Also at this time, the British were busy competing with the other European countries colonizing Africa. Colonialism was even better than slavery. Instead of the hassle of shipping slaves across the ocean, they could just enslave the people at home. Ten-million Congolese lost their lives in forced labor, harvesting natural rubber for King Leopold of Belgium. Cecil Rhodes (started De Beers and the Rhodes Scholarship) founded Rhodesia, and wanted to see an all British Africa. The European nations had a meeting in Berlin and decided to divide up the entire continent of Africa amongst themselves.
So, as Larry Gonick points out, the start of the century that saw the end of slavery, saw the beginning of colonialism in Africa.