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.

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.

The Dilettante on the History of Slavery and Colonialism in the Modern World

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.