Sums of squares, the octonions, and (7,3,1) - MAA Sections

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. . . becomes the incidence matrix for (7, 3, 1) 1 2 3 4 5 6 7 B 1 1 1 0 1 0 0 0 124 B 2 0 1 1 0 1 0 0 235 B 3 0 0 1 1 0 1 0 346 B 4 0 0 0 1 1 0 1 457 B 5 1 0 0 0 1 1 0 561 B 6 0 1 0 0 0 1 1 672 B 7 1 0 1 0 0 0 1 713 I told you it was important. Brown The Octonions
Octonion multiplication, explained The following are the seven blocks of the (7, 3, 1) design, with the given internal orderings: (1, 2, 4), (2, 3, 5), (3, 4, 6), (4, 5, 7), (5, 6, 1), (6, 7, 2), and (7, 1, 3). For distinct i, j ∈ {1, 2, 3, 4, 5, 6, 7}, define e i e j = e k = −e j e i , where (i, j, k) is the unique block containing i and j, in the given internal ordering. What is e 4 e 6 ? The relevant block is (3, 4, 6), so e 4 e 6 = e 3 . What is e 5 e 1 ? The relevant block is (5, 6, 1), so e 5 e 1 = −e 1 e 5 = −e 6 . That’s why “the octonion units” is a name for (7, 3, 1). But wait — there’s another reason. Brown The Octonions
Page 1 and 2: Sums of squares, the octonions, and
Page 3 and 4: A question about sums of squares Su
Page 5 and 6: The (7, 3, 1) block design The (7,
Page 7 and 8: One and two squares One square The
Page 9 and 10: Eight squares (a1 2 + a2 2 + a2 3 +
Page 11 and 12: Multiplying the octonion units ∗
Page 13: The matrix, altered slightly . . .
Page 17: THANK YOU! Brown The Octonions

Sums of squares, the octonions, and (7,3,1) - MAA Sections

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