Odd version of 24 squares formula?
The Leech lattice is related to the formula:
$$1^2+2^2+3^2+....+24^2=70^2$$
This is related in turn to 26D bosonic string theory.
Is there another known formula that involves the integers 1..8, which is related perhaps to $E_8$ or 10D supergravity?
At first I thought it might be the sums of squares of the first 8 odd numbers but this didn't give anything interesting. (Because supersymmetry includes half spin particles $1/2,3/2,5/2$ etc.
(The Leech lattice and $E_8$ being the lattices with maximal sphere packing in their respective dimensions).
Edit: I found another examples. Not sure if they have any relevance:
$$2^2+5^2+8^2+11^2+14^2+17^2+20^2+23^2+26^2=48^2$$
Which goes up in 3's and also this is 9 terms. Probably just a coincidence.
spheres integer-lattices
asked Dec 9 at 18:52
zooby
971616
add a comment |
The Leech lattice is related to the formula:
$$1^2+2^2+3^2+....+24^2=70^2$$
This is related in turn to 26D bosonic string theory.
Is there another known formula that involves the integers 1..8, which is related perhaps to $E_8$ or 10D supergravity?
At first I thought it might be the sums of squares of the first 8 odd numbers but this didn't give anything interesting. (Because supersymmetry includes half spin particles $1/2,3/2,5/2$ etc.
(The Leech lattice and $E_8$ being the lattices with maximal sphere packing in their respective dimensions).
Edit: I found another examples. Not sure if they have any relevance:
$$2^2+5^2+8^2+11^2+14^2+17^2+20^2+23^2+26^2=48^2$$
Which goes up in 3's and also this is 9 terms. Probably just a coincidence.
spheres integer-lattices
asked Dec 9 at 18:52
zooby
971616
Check the eight squares identity en.wikipedia.org/wiki/Degen%27s_eight-square_identity and this question math.stackexchange.com/questions/651080/…
– John Wayland Bales
Dec 9 at 19:32
Interesting. This makes me wonder if there is a 24-square identity!
– zooby
Dec 10 at 3:41
There is the Pfister 16 square identity en.wikipedia.org/wiki/Pfister%27s_sixteen-square_identity and a free paper on the 2-, 4-, and 8-square identities en.wikipedia.org/wiki/Pfister%27s_sixteen-square_identity. I do not know of a 24-square identity.
– John Wayland Bales
Dec 10 at 3:55
add a comment |
The Leech lattice is related to the formula:
$$1^2+2^2+3^2+....+24^2=70^2$$
This is related in turn to 26D bosonic string theory.
Is there another known formula that involves the integers 1..8, which is related perhaps to $E_8$ or 10D supergravity?
At first I thought it might be the sums of squares of the first 8 odd numbers but this didn't give anything interesting. (Because supersymmetry includes half spin particles $1/2,3/2,5/2$ etc.
(The Leech lattice and $E_8$ being the lattices with maximal sphere packing in their respective dimensions).
Edit: I found another examples. Not sure if they have any relevance:
$$2^2+5^2+8^2+11^2+14^2+17^2+20^2+23^2+26^2=48^2$$
Which goes up in 3's and also this is 9 terms. Probably just a coincidence.
spheres integer-lattices
asked Dec 9 at 18:52
zooby
971616
The Leech lattice is related to the formula:
$$1^2+2^2+3^2+....+24^2=70^2$$
This is related in turn to 26D bosonic string theory.
Is there another known formula that involves the integers 1..8, which is related perhaps to $E_8$ or 10D supergravity?
At first I thought it might be the sums of squares of the first 8 odd numbers but this didn't give anything interesting. (Because supersymmetry includes half spin particles $1/2,3/2,5/2$ etc.
(The Leech lattice and $E_8$ being the lattices with maximal sphere packing in their respective dimensions).
Edit: I found another examples. Not sure if they have any relevance:
$$2^2+5^2+8^2+11^2+14^2+17^2+20^2+23^2+26^2=48^2$$
Which goes up in 3's and also this is 9 terms. Probably just a coincidence.
spheres integer-lattices
spheres integer-lattices
asked Dec 9 at 18:52
zooby
971616
asked Dec 9 at 18:52
zooby
971616
edited Dec 10 at 3:55
asked Dec 9 at 18:52
zooby
971616
asked Dec 9 at 18:52
zooby
971616
asked Dec 9 at 18:52
zooby
971616
971616
Check the eight squares identity en.wikipedia.org/wiki/Degen%27s_eight-square_identity and this question math.stackexchange.com/questions/651080/…
– John Wayland Bales
Dec 9 at 19:32
Interesting. This makes me wonder if there is a 24-square identity!
– zooby
Dec 10 at 3:41
There is the Pfister 16 square identity en.wikipedia.org/wiki/Pfister%27s_sixteen-square_identity and a free paper on the 2-, 4-, and 8-square identities en.wikipedia.org/wiki/Pfister%27s_sixteen-square_identity. I do not know of a 24-square identity.
– John Wayland Bales
Dec 10 at 3:55
add a comment |
Check the eight squares identity en.wikipedia.org/wiki/Degen%27s_eight-square_identity and this question math.stackexchange.com/questions/651080/…
– John Wayland Bales
Dec 9 at 19:32
Interesting. This makes me wonder if there is a 24-square identity!
– zooby
Dec 10 at 3:41
There is the Pfister 16 square identity en.wikipedia.org/wiki/Pfister%27s_sixteen-square_identity and a free paper on the 2-, 4-, and 8-square identities en.wikipedia.org/wiki/Pfister%27s_sixteen-square_identity. I do not know of a 24-square identity.
– John Wayland Bales
Dec 10 at 3:55
Check the eight squares identity en.wikipedia.org/wiki/Degen%27s_eight-square_identity and this question math.stackexchange.com/questions/651080/…
– John Wayland Bales
Dec 9 at 19:32
Check the eight squares identity en.wikipedia.org/wiki/Degen%27s_eight-square_identity and this question math.stackexchange.com/questions/651080/…
– John Wayland Bales
Dec 9 at 19:32
Interesting. This makes me wonder if there is a 24-square identity!
– zooby
Dec 10 at 3:41
Interesting. This makes me wonder if there is a 24-square identity!
– zooby
Dec 10 at 3:41
There is the Pfister 16 square identity en.wikipedia.org/wiki/Pfister%27s_sixteen-square_identity and a free paper on the 2-, 4-, and 8-square identities en.wikipedia.org/wiki/Pfister%27s_sixteen-square_identity. I do not know of a 24-square identity.
– John Wayland Bales
Dec 10 at 3:55
There is the Pfister 16 square identity en.wikipedia.org/wiki/Pfister%27s_sixteen-square_identity and a free paper on the 2-, 4-, and 8-square identities en.wikipedia.org/wiki/Pfister%27s_sixteen-square_identity. I do not know of a 24-square identity.
– John Wayland Bales
Dec 10 at 3:55
add a comment |
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Check the eight squares identity en.wikipedia.org/wiki/Degen%27s_eight-square_identity and this question math.stackexchange.com/questions/651080/…
– John Wayland Bales
Dec 9 at 19:32
Interesting. This makes me wonder if there is a 24-square identity!
– zooby
Dec 10 at 3:41
There is the Pfister 16 square identity en.wikipedia.org/wiki/Pfister%27s_sixteen-square_identity and a free paper on the 2-, 4-, and 8-square identities en.wikipedia.org/wiki/Pfister%27s_sixteen-square_identity. I do not know of a 24-square identity.
– John Wayland Bales
Dec 10 at 3:55