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If you’re in Australia, there’s sure to be something on
https://www. sbs.com.au/ondemand/
that will grab you. “Atlanta”, “The Good Fight”, “Halt and Catch Fire” (until 30 Nov), “My Brilliant Friend”. I plan to watch the movie “Drive My Car” on the w/end. And the ads are so repetitive they no longer register; when you’ve seen them all 500 times, your mind just filters them out, like the refrigerator motor switching on and off.

# orderedItems[].object.contentMap.en
If you’re in Australia, there’s sure to be something on
https://www. sbs.com.au/ondemand/
that will grab you. “Atlanta”, “The Good Fight”, “Halt and Catch Fire” (until 30 Nov), “My Brilliant Friend”. I plan to watch the movie “Drive My Car” on the w/end. And the ads are so repetitive they no longer register; when you’ve seen them all 500 times, your mind just filters them out, like the refrigerator motor switching on and off.

# orderedItems[].object.content
One thing I find hilarious in “Halt and Catch Fire” is when people are writing code, typing rapidly, and the result appears on their screen at an absolutely uniform rate, as if, umm, a piece of visual effects software was disclosing pre-existing text at a metronomic 100ms/char.

# orderedItems[].object.contentMap.en
One thing I find hilarious in “Halt and Catch Fire” is when people are writing code, typing rapidly, and the result appears on their screen at an absolutely uniform rate, as if, umm, a piece of visual effects software was disclosing pre-existing text at a metronomic 100ms/char.

# orderedItems[].object.content
How erudite of my spellchecker not to redline “throughly”, which is apparently an archaic version of “thoroughly” that isn’t mentioned in any dictionary I own, and couldn’t possibly be, you know, a typo for “thoroughly”. Fck.

# orderedItems[].object.contentMap.en
How erudite of my spellchecker not to redline “throughly”, which is apparently an archaic version of “thoroughly” that isn’t mentioned in any dictionary I own, and couldn’t possibly be, you know, a typo for “thoroughly”. Fck.

# orderedItems[].object.attachment[].name
How erudite of my spellchecker not to redline “throughly”, which is apparently an archaic version of “thoroughly” that isn’t mentioned in any dictionary I own, and couldn’t possibly be, you know, a typo for “thoroughly”. Fck.

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@johncarlosbaez No, it looks the correct size.

# orderedItems[].object.contentMap.en
@johncarlosbaez No, it looks the correct size.

# orderedItems[].object.content
@johncarlosbaez I deleted them from my web site because I forgot I’d linked to them externally (they weren’t linked to by any pages on my site). But I’ve restored them from backups now.

# orderedItems[].object.contentMap.en
@johncarlosbaez I deleted them from my web site because I forgot I’d linked to them externally (they weren’t linked to by any pages on my site). But I’ve restored them from backups now.

# orderedItems[].object.content
@johncarlosbaez I deleted them from my web site because I forgot I’d linked to them externally (they weren’t linked to by any pages on my site). But I’ve restored them from backups now.

# orderedItems[].object.contentMap.en
@johncarlosbaez I deleted them from my web site because I forgot I’d linked to them externally (they weren’t linked to by any pages on my site). But I’ve restored them from backups now.

# orderedItems[].object.content
I’m just waiting for someone to train an LLM exclusively on a religion’s sacred texts, and then market it as a way to converse with the associated deity or deities.

# orderedItems[].object.contentMap.en
I’m just waiting for someone to train an LLM exclusively on a religion’s sacred texts, and then market it as a way to converse with the associated deity or deities.

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An ingenious way to judge an animal’s expectations of how things in the world behave: see which sleight-of-hand tricks succeed in fooling it.

https://theconversation.com/we-perform ed-magic-tricks-on-birds-to-see-how-they-perceive-the-world-161772

# orderedItems[].object.contentMap.en
An ingenious way to judge an animal’s expectations of how things in the world behave: see which sleight-of-hand tricks succeed in fooling it.

https://theconversation.com/we-perform ed-magic-tricks-on-birds-to-see-how-they-perceive-the-world-161772

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In our grid above, every row and every column has i,j,k appear as the second quaternion in the pair. So any q that is a *simultaneous* eigenvector of all 3 rotations in a row or a column will map i,j and k to ± the first quaternion in the pair, which is always one of i,j,k.

So q is a symmetry of the octahedron.

# orderedItems[].object.contentMap.en
In our grid above, every row and every column has i,j,k appear as the second quaternion in the pair. So any q that is a *simultaneous* eigenvector of all 3 rotations in a row or a column will map i,j and k to ± the first quaternion in the pair, which is always one of i,j,k.

So q is a symmetry of the octahedron.

# orderedItems[].object.content
… then those quaternion pairs yield the same matrices as the Mermin grid!

Now, suppose q is an eigenvector of the rotation from the pair (r,s), with eigenvalue ±1. Then:

r q s^{–1} = ± q
r q = ± q s
r = ± q s q^{–1}
Rot_q(s) = ± r

This means construing q as a three-dimensional rotation, it will map s to ± r.

# orderedItems[].object.contentMap.en
… then those quaternion pairs yield the same matrices as the Mermin grid!

Now, suppose q is an eigenvector of the rotation from the pair (r,s), with eigenvalue ±1. Then:

r q s^{–1} = ± q
r q = ± q s
r = ± q s q^{–1}
Rot_q(s) = ± r

This means construing q as a three-dimensional rotation, it will map s to ± r.

# orderedItems[].object.content
What else consists of two 24-cells in R^4?

The binary octahedral group, containing the 48 quaternions q, that when construed as rotations via:

x → q x q^{-1}

give symmetries of the octahedron with vertices ±i, ±j, ±k.

So … how can we link the Mermin grid to that group?

Consider the 3 × 3 grid of pairs of quaternions:

(j, j) (i, i) (k, k)
(i, k) (k, j) (j, i)
(k, i) (j, k) (i, j)

If we treat a pair of quaternions (r,s) as defining a rotation in R^4 via:

x → r x s^{-1}

# orderedItems[].object.contentMap.en
What else consists of two 24-cells in R^4?

The binary octahedral group, containing the 48 quaternions q, that when construed as rotations via:

x → q x q^{-1}

give symmetries of the octahedron with vertices ±i, ±j, ±k.

So … how can we link the Mermin grid to that group?

Consider the 3 × 3 grid of pairs of quaternions:

(j, j) (i, i) (k, k)
(i, k) (k, j) (j, i)
(k, i) (j, k) (i, j)

If we treat a pair of quaternions (r,s) as defining a rotation in R^4 via:

x → r x s^{-1}

# orderedItems[].object.content
@johncarlosbaez Not a coincidence, it’s exactly the same thing! (Though I mislaid that animation and redid it here, so it might not be visually identical)

# orderedItems[].object.contentMap.en
@johncarlosbaez Not a coincidence, it’s exactly the same thing! (Though I mislaid that animation and redid it here, so it might not be visually identical)

# orderedItems[].object.content
If you take the 3 operators in any row of the Mermin grid, there are 4 orthogonal vectors that are simultaneously eigenvectors of all three.

These 12 vectors and their opposites are the vertices of a 24-cell! And if you do the same for the columns, you get the 24-cell dual to it (whose vertices are normals to the 24 octahedral cells of the first 24-cell).

In the image below, the vertices of each colour are simultaneous eigenvectors for each of the three rows of operators.

# orderedItems[].object.contentMap.en
If you take the 3 operators in any row of the Mermin grid, there are 4 orthogonal vectors that are simultaneously eigenvectors of all three.

These 12 vectors and their opposites are the vertices of a 24-cell! And if you do the same for the columns, you get the 24-cell dual to it (whose vertices are normals to the 24 octahedral cells of the first 24-cell).

In the image below, the vertices of each colour are simultaneous eigenvectors for each of the three rows of operators.