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Les Éléments de l'Art Arabe,

Lots of plates of pretty geometric girih patterns, from an 1879 book by Jules Bourgoin. Via a comment at

ε² = 0

It makes sense to think of ε as a first approximation for infinitesimals: smaller than any non-infinitesimal, but still greater than zero.

You might also notice that Grassmann's exterior algebra works in a similar way.

Grassmann's exterior algebra also appears as the wedge product in differential algebra. That smells like a kind of unification of the two interpretations above, but I don't know how to make that precise...

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Recall that quarternions are like complex numbers on steroids. They have three imaginary units i, j, k each of which squares to minus one, just like the complex unit:

i² = -1

To get dual quarternions, take two quarternions A and B and think of one of them as carrying a new kind of symbol ε:


ε also reacts in a funny way when squared:

ε² = 0


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Dual quarternions are an 8-dimensional algebra that can be used to represent not only rotation and scaling, but also translations in R³.

Using mere quarternions for rotations in R³ is a well-known trick. But translations work differently, so we need a new kind of , octonions simply won't work here.


How many years until General Direct Dialed telephony dies?


Archimedes came up with a number system that could count up to 10^(8*10^16), and after summarizing the history of measuring the size of our cosmos, Parallax Nick asks "what was he hoping to count with that number?"

The Planck Reckoner

One or two bugs aside, I really like Nick's style, and the format, too! Go and click through to his channel, and watch all of his series about water in the solar system!

@11011110 Things get particularly weird (and wonderful) when your fractional base is smaller than 1. You can see a problem about inserting things into a rotating queue that leads to these kinds of expansions in these slides:

#fractals #graphics #monochrome #art

An excerpt from the parameter plane of this formula iterated in a loop (p = 2):

z := z^p + c
z := (|x| - i y)^p + c
z := (x - i |y|)^p + c

Apparently some people are only just discovering that AI is interdisciplinary. When I first got interested in it it was grouped in with philosophy and psychology.

req is a text snippet router: given a string, say from your X11 cut buffer, it will classify and extract information contents, and merge that with context information. You write rule files ("ports") in awk to match against this data and to generate a menu of available shell commands.

For @dredmorbius . Feel free to bug me about anything.

I'm indebted to @johncarlosbaez for his talk about the relation of G2 to the octonions, from which I learned all I know about G2. Find out more here:

Thanks, John!

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Also funny: since the big ball isn't really a ball, but a projective plane, that that also played a role in my construction of SO(3) above. No rotation group corresponds to RP² it somehow seems to sit between SO(2) and SO(3)!

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Fun fact about G2: it's 5-dimensional, but the little ball has only two dimensions of infinitesimal freedom! You could get to any 5d point by navigating as if you were walking on a 2d surface! Orienting a ball by rolling it on the ground is similar...

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This spinor rolling on a projective plane, where the latter has three times the spinor's radius, models the real valued lie algebra g2. However, this also works for complex values. This is what Cartan originally described.

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So, let's solve the puzzle: Start at the center s of our, say, white hemiball. After a quarter turn we get to -s the center of the black hemiball. Now all directions point back at s! Any other quarter turn leads back!

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Take another copy of SO(3) also looking like the inside of a ball, and glue that onto our original along their spherical boundaries. Leaving a ball turns you over, but reentering turns you upright again!

No matter how you cross Spin(3) you never get back to your starting point turned upside down. The twist is gone! This pair of hemiballs is called 3-sphere or S³. Psst, this object actually lives in ! Now you know a bit more about how that works!

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Picking a pair of antipodal points alone leads to a projective plane RP(2). As flipping the axis can be compensated by rotating in the opposite direction.

When you fibrate SO(3) with RP² as base space like this, you get full 360° circles as fibers. Think of them as the diameters of the ball.

Now let's double cover that beast:

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Spinors are double covers for rotation groups. Double covering sounds like it would become more complicated. But on the contrary, it really makes things simpler: Take SO(3): the space of rotations in 3-space! Pick an axis of a sphere: a pair of antipodal points. We can then rotate around that axis. The result is a closed 3-space.

It looks like a ball where, if you try to leave through the wall, you reappear on the opposite side, turned upside down! Not mirrored, turned.

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Puzzle: Starting at a pole, roll an electron around our 3 times bigger ball. After a quarter turn its orientation became the negative –s of our starting orientation s. Can we get back to s by rolling along the equator?

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Dirac's belt trick shows how to undo double twists without moving the electron. Or the ground. It also works with many ribbons at once! And no matter what the rotation axis was! That's a spinor! You need two full turns to get the identity.

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