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

Sums of squares, the octonions, and (7, 3, 1)
Ezra Brown
Virginia Tech
MD/DC/VA Spring Section Meeting
Stevenson University
April 14, 2012
Brown
The Octonions

What To Expect
Sums of squares
Normed algebras
(7, 3, 1)
1-, 2- and 4-square identities and their algebras
8 squares
The octonions
The connection with (7, 3, 1)
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The Octonions

A question about sums of squares
Sums of squares
For which n can the product of two sums of n squares always be
written as a sum of n squares?
Answer (A. Hurwitz, 1898): For n = 1, 2, 4 and 8 — and for no other
positive integers.
Theorem: Each sums-of-squares identity is associated with a normed
algebra over the real numbers.
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The Octonions

Normed algebras
A real algebra is a vector space over R that has a vector
multiplication that distributes over vector addition.
A normed algebra is a real algebra A equipped with a mapping
N : A → R such that N(uv) = N(u)N(v) for all u, v ∈ A.
The real numbers R and the complex numbers C both have such
norms.
We’ll meet the other two shortly.
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The Octonions

The (7, 3, 1) block design
The (7, 3, 1) block design is:
B 1 124
B 2 235
B 3 346
B 4 457
B 5 561
B 6 672
B 7 713
a set V of 7 items, and a collection of 7 subsets of V called blocks,
such that
each block contains three items,
each item is in three blocks, and
each pair of items is in exactly one block together.
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The Octonions

The incidence matrix for (7, 3, 1)
The incidence matrix for (7, 3, 1) is the 7 × 7 (0, 1) matrix M with
M ij = 1 if and only if B i contains the jth object:
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
Remember this, because it’s important.
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The Octonions

One and two squares
One square
The identity: a 2 · b 2 = (ab) 2
The algebra: R, the real numbers
The norm: N(r) = r 2 .
Two squares
The identity: (a 2 + b 2 )(c 2 + d 2 ) = (ac − bd) 2 + (ad + bc) 2 , due to
Diophantus (3rd century) and Brahmagupta (7th century)
The algebra: C = {a + bi|a, b ∈ R, i 2 = −1}, the complex numbers
The norm: N(a + bi) = a 2 + b 2
The origins of C: solution of the cubic, differential equations, the
geometric complex plane
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The Octonions

Four squares
The identity, due to Euler (1748):
(a 2 1 + a 2 2 + a 2 3 + a 2 4)(b 2 1 + b 2 2 + b 2 3 + b 2 4) =
= (a 1 b 1 − a 2 b 2 − a 3 b 3 − a 4 b 4 ) 2 + (a 1 b 2 + a 2 b 1 + a 3 b 4 − a 4 b 3 ) 2
+(a 1 b 3 − a 2 b 4 + a 3 b 1 + a 4 b 2 ) 2 + (a 1 b 4 + a 2 b 3 − a 3 b 2 + a 4 b 1 ) 2
The algebra: the quaternions
H = {a 1 + a 2 i + a 3 j + a 4 k|a 1 , a 2 , a 3 , a 4 ∈ R, i 2 = j 2 = k 2 = ijk = −1}
The norm: N(a 1 + a 2 i + a 3 j + a 4 k) = a 2 1 + a2 2 + a2 3 + a2 4
The origins: W. R. Hamilton, who —
failed in an attempt to define a multiplication on R 3 , and then —
succeeded in defining a noncommutative multiplication on R 4 .
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The Octonions

Eight squares
(a1 2 + a2 2 + a2 3 + a2 4 + a2 5 + a2 6 + a2 7 + a2 8 )
×(b1 2 + b2 2 + b2 3 + b2 4 + b2 5 + b2 6 + b2 7 + b2 8 )
= (a 1 b 1 − a 2 b 2 − a 3 b 3 − a 4 b 4 − a 5 b 5 − a 6 b 6 − a 7 b 7 − a 8 b 8 ) 2
+(a 1 b 2 + a 2 b 1 + a 3 b 4 − a 4 b 3 + a 5 b 6 − a 6 b 5 − a 7 b 8 + a 8 b 7 ) 2
+(a 1 b 3 − a 2 b 4 + a 3 b 1 + a 4 b 2 + a 5 b 7 + a 6 b 8 − a 7 b 5 − a 8 b 6 ) 2
+(a 1 b 4 + a 2 b 3 − a 3 b 2 + a 4 b 1 + a 5 b 8 − a 6 b 7 + a 7 b 6 − a 8 b 5 ) 2
+(a 1 b 5 − a 2 b 6 − a 3 b 7 − a 4 b 8 + a 5 b 1 + a 6 b 2 + a 7 b 3 + a 8 b 4 ) 2
+(a 1 b 6 + a 2 b 5 − a 3 b 8 + a 4 b 7 − a 5 b 2 + a 6 b 1 − a 7 b 4 + a 8 b 3 ) 2
+(a 1 b 7 + a 2 b 8 + a 3 b 5 − a 4 b 6 − a 5 b 3 + a 6 b 4 + a 7 b 1 − a 8 b 2 ) 2
+(a 1 b 8 − a 2 b 7 − a 3 b 6 + a 4 b 5 − a 5 b 4 − a 6 b 3 + a 7 b 2 + a 8 b 1 ) 2
Due to F. Degen (1818), J. T. Graves (1843) and A. Cayley (1845)
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The Octonions

Eight squares, continued
The algebra: the octonions O, where
O = {a 0 + a 1 e 1 + . . . + a 7 e 7 |a 0 , . . . , a 7 ∈ R, e 2 t = −1}; the e t are
called the octonion units
The norm:
N(a 0 + a 1 e 1 + · · · + a 7 e 7 ) = a 2 1 + a2 1 + a2 2 + a2 3 + a2 4 + a2 5 + a2 6 + a2 7
The origins: J. T. Graves, who —
received Hamilton’s letter announcing the quaternions, and then —
went him one better by defining a noncommutative and nonassociative
multiplication on R 8 .
The multiplication is given by the following table.
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The Octonions

Multiplying the octonion units
∗ 1 e 1 e 2 e 3 e 4 e 5 e 6 e 7
1 1 e 1 e 2 e 3 e 4 e 5 e 6 e 7
e 1 e 1 −1 e 4 e 7 −e 2 e 6 −e 5 −e 3
e 2 e 2 −e 4 −1 e 5 e 1 −e 3 e 7 −e 6
e 3 e 3 −e 7 −e 5 −1 e 6 e 2 −e 4 e 1
e 4 e 4 e 2 −e 1 −e 6 −1 e 7 e 3 −e 5
e 5 e 5 −e 6 e 3 −e 2 −e 7 −1 e 1 e 4
e 6 e 6 e 5 −e 7 e 4 −e 3 −e 1 −1 e 2
e 7 e 7 e 3 e 6 −e 1 e 5 −e 4 −e 2 −1
This looks very strange . . .
Multiplication table for the octonion units.
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The Octonions

The table as a matrix of signs of the e i
. . . but looks are deceptive.
⎡
H =
⎢
⎣
1 1 1 1 1 1 1 1
1 −1 1 1 −1 1 −1 −1
1 −1 −1 1 1 −1 1 −1
1 −1 −1 −1 1 1 −1 1
1 1 −1 −1 −1 1 1 −1
1 −1 1 −1 −1 −1 1 1
1 1 −1 1 −1 −1 −1 1
1 1 1 −1 1 −1 −1 −1
⎤
⎥
⎦
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The Octonions

The matrix, altered slightly . . .
Remove the top row and left column, replace each −1 by a zero, and
move the bottom row to the top. Here’s the result:
⎡
⎤
1 1 0 1 0 0 0
0 1 1 0 1 0 0
0 0 1 1 0 1 0
0 0 0 1 1 0 1
⎢ 1 0 0 0 1 1 0
⎥
⎣ 0 1 0 0 0 1 1 ⎦
1 0 1 0 0 0 1
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The Octonions

. . . 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.
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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.
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The Octonions

O is another name for (7, 3, 1)
Seven quaternion subalgebras of O
O contains seven complex subalgebras C n = R〈e n 〉 and seven
quaternion subalgebras H n = R〈e t , e u , e v 〉, where {t, u, v} is a block
in (7, 3, 1)
Each H n contains three of the C k .
Each C k is contained in three of the H n .
Each pair {C k , C m } is contained in a unique H n together.
The (7, 3, 1) design sits inside O
The above design of subalgebras of O has the same incidence matrix
as (7, 3, 1).
The two designs are the same!
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The Octonions

THANK YOU!
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The Octonions