The tesseract entered popular culture through Madeleine L’Engle’s “A Wrinkle in Time”, though L’Engle caused some of her readers confusion when one of the characters in “A Wrinkle in Time”, the prodigy Charles Wallace Murray, declared “Well, the fifth dimension’s a tesseract.” L’Engle wasn’t sure how to reconcile Hinton’s ideas about the fourth dimension with Einstein’s, so she put Hinton’s fourth dimension after Einstein’s, demoting it from fourth place to fifth.

]]>So thanks!

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Note in particular Fig 1:

This is a sequence of sections of a tesseract, going along a diagonal between opposite vertices. Which bears a striking resemblance to the tetrahedron family from Part 7 of this series.

In a similar vein, the 24-cell (ICO) according to this site:

http://www.polytope.net/hedrondude/regulars.htm

has this sequence of sections (from one of the cells this time):

which appears to be the octahedron through cuboctahedron of the cube family from Part 5.

Coincidence?

]]>Boole had 5 daughters, the 5th being Ethel Lilian, who married Wilfrid Michael Voynich:

]]>I want to know the pendulum’s frequency, because I understand that there is a relationship with sound frequencies, and apparently the harmonograph can be set to mimic the frequencies of different musical intervals – see “Harmonograph: A visual guide to the mathematics of music” by Anthony Ashton, Wooden Books.

]]>I just saw your harmonograph. It’s beautiful!

Anita replied:

Thank you John – I am still laboring with it in the metal workshop!

Oh, so the gadget in the picture above is not the harmonograph? I guess that makes sense—I don’t see a pendulum. What is that gadget? It’s beautiful…

I know that there is a mathematical formula for the pendulum frequencies, but as a non-mathematician it still defeats me – have you written about this anywhere?

It depends what you mean. There’s a simple approximate formula that’s good if the pendulum doesn’t swing too high:

Here is the period of the pendulum, is its length, and is the acceleration of gravity, about 9.807 meters/second^{2}.

However, if the pendulum swings high, its period becomes longer, as you can easily see if you imagine it swinging *very* high, like a child trying to swing all the way over the swing set. We need a more complicated formula to describe this. The exact formula involves elliptic integrals, and I’ve written about that… but if your pendulum doesn’t swing *too* high, you may be content with this formula: not the first one, but the one with correction terms involving which is the maximum angle from vertical that the pendulum swings.

When is 23°, the simple formula I gave above is wrong by 1%. When is 30° the pendulum looks like this:

and when is 120° the period is noticeably longer:

I suspect that your harmonograph will be more complicated. It’s not just swinging back and forth in one direction: it may be a true ‘spherical pendulum’, or maybe something even more complicated. But if it doesn’t swing too high, its period should still be given approximately by the simple formula I wrote down.

Why do you want to know the pendulum’s frequency?

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