Part 9 of 15. The octonions' 49 imaginary products are a single geometric picture. Then parentheses start to matter, Hurwitz proves the doubling must stop, and the unit octonions become the last sphere you can multiply on.
The Cayley-Dickson recipe took us to 8D: one real part (e0 = 1) and seven imaginaries e1, …, e7, leaving 49 products to pin down. The Fano plane, the smallest projective plane with seven points and seven lines, holds the entire multiplication pattern in one picture.
Three rules, and that is all.
ei, ej lie on exactly one line of the plane. (Every line is a triple of units, and every line is a copy of the quaternions.)ei · ej = +ek. Reversing the cycle flips the sign: ej · ei = −ek.ei · ei = −1. (The plane doesn't show this, it's the same rule as the quaternions.)Figure 1. The Fano plane. Seven imaginary units sit at the seven points; seven lines (three sides, three altitudes, and one inner circle) carry the multiplication rules. Every line is a cyclic quaternion triple (ei, ej, ek) with ei · ej = ek, ej · ek = ei, and ek · ei = ej. Reversing any of those flips the sign.
| × | e₁ | e₂ | e₃ | e₄ | e₅ | e₆ | e₇ |
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Figure 2. The octonion multiplication table for the imaginary units. The diagonal is −1 (every imaginary squares to minus one). The table is antisymmetric across the diagonal: transposing any cell flips its sign. Positive products are green; negative ones are red. Hovering a cell loads the corresponding pair into the explorer above.
The Fano plane is a statement of symmetry: every imaginary direction plays the same role, every line is a quaternion sub-copy, and the 42 ordered imaginary pairs each sit on exactly one line. No direction is privileged. But multiplying three units breaks the clean story.
For reals, complexes, and quaternions, $(ab)c = a(bc)$. For octonions, associativity breaks.
This is why octonions resist matrix representation: matrix multiplication is always associative.
The associator measures the failure: $[a, b, c] = (ab)c - a(bc)$, often nonzero in 8D.
Octonions aren't always non-associative: units on the same Fano line behave. Find a triple $(a, b, c)$ that associates, then one that diverges.
Figure 3. The associative tree. The blue path computes $(ab)c$. The red path computes $a(bc)$. If the nodes match, the units are associative. If they land on different nodes, the algebra has diverged.
Any two octonions generate a quaternion sub-copy (diassociativity). With just two units, parentheses move freely. Non-associativity needs three directions not on the same Fano line.
Cayley-Dickson built reals, complexes, quaternions, octonions. In 1898 Hurwitz proved this is the limit: octonions are the last algebra preserving $|ab| = |a||b|$.
Doubling to 16D gives the sedenions, where multiplication and distance decouple.
In a number system, $ab = 0$ usually forces $a = 0$ or $b = 0$. In 16D you can find nonzero zero-divisors. Find one.
Figure 4. The 16D minefield. In 1, 2, 4, or 8 dimensions, you can't hit zero unless an input is zero. In 16D, the "holes" in the algebra appear.
Consequence: zero-divisors mean division loses uniqueness. The sedenions are an algebra but not a number system.
Each Cayley-Dickson doubling sacrifices one algebraic property.
| Algebra | Ordered | Commutative | Associative | Alternative | Normed |
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Figure 5. The property-loss staircase. Each doubling from $\mathbb{R}$ to $\mathbb{S}$ trades one algebraic property for double the dimensions. The octonions sit at the last step where multiplication still respects geometry.
Without associativity, parentheses no longer move freely. But octonions obey survivor laws: the Moufang identities, defining a Moufang loop.
The four Moufang identities all say: wrapping a product with the same element in a fixed pattern cancels the non-associativity. Weaker than associativity, strong enough for real algebra.
The four Moufang laws:
1. $a(b(ac)) = ((ab)a)c$ — left Moufang
2. $((ca)b)a = c(a(ba))$ — right Moufang
3. $(ab)(ca) = a(bc)a$ — middle Moufang
4. $(ab)(ca) = (a(bc))a$ — flexible variant
The element $a$ always appears on both sides, forcing the non-associativity to cancel.
Figure 6. Moufang identity verifier. Choose among the four classic Moufang laws and test them with random elements. In 8D (octonions) they always hold. Toggle to 16D (sedenions) and watch them fail — the Cayley-Dickson construction past the octonions breaks these identities, just as Hurwitz's theorem predicts.
The unit elements of the four division algebras inherit multiplication: $|ab| = |a||b|$ means a product of unit vectors is a unit vector. So $S^0, S^1, S^3, S^7$ are manifolds with multiplication.
These four are also the only parallelizable spheres: a smooth global frame of independent tangent vectors exists everywhere, with no "bald spots."
Figure 7. The four parallelizable spheres. Each is the unit sphere of a normed division algebra. The number of independent tangent vector fields equals the dimension minus one. No other spheres have this property.
You are a pilot on $S^7$. Each button multiplies your 8D unit vector by an imaginary unit $e_k$, rotating along a great circle. Reach the target.
Figure 8. Navigating the 7-sphere. Every multiplication moves you along a great circle. On $S^7$ you can never get stuck — the seven independent tangent directions guarantee there is always a clear way to fly.
The cliff. Hurwitz stops at 8D: no $S^{15}$ or $S^{31}$ has this navigation. $S^7$ is the largest parallelizable sphere.
Inside $\mathbb{O}$ sits a maximal integral order (a ring of integer octonions closed under multiplication) with exactly 240 units. Those 240 units are point-for-point the 240 roots of $E_8$. The non-associative 8D structure of this article is the same object as the densest known 8D sphere packing.
The 240 unit integer octonions split cleanly by their real (scalar) part:
Figure 9. The 240 unit integer octonions partitioned by scalar part. Each of the 240 dots is one unit; the three bands are the three scalar classes. Click any band to see a breakdown of its members with specific coordinate examples. The same 240 drawn on the Coxeter mandala of $E_8$, coloured instead by which of the ten 24-cells they inhabit, is explainer 13.
The exceptional series is octonionic. $\mathfrak{g}_2 = \mathrm{Aut}(\mathbb{O})$; $\mathfrak{f}_4, \mathfrak{e}_6, \mathfrak{e}_7, \mathfrak{e}_8$ fill the Freudenthal magic square (explainer 10). $E_8$'s 240 roots are the visible surface of integer octonions.
The Fano plane stores all 49 imaginary products as seven oriented lines; each line is a quaternion sub-copy.
The associator $[a,b,c] = (ab)c - a(bc)$ measures how badly associativity fails. Zero on a Fano line, nonzero otherwise.
Diassociativity: any two octonions generate a copy of $\mathbb{H}$. Non-associativity only bites with three independent directions.
Hurwitz's theorem (1898): $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$, $\mathbb{O}$ are the only normed division algebras. In 16D, zero divisors appear and $|ab| = |a||b|$ fails.
The Moufang identities survive the doubling to 8D and make the unit octonions into a Moufang loop.
$S^0, S^1, S^3, S^7$ are the only parallelizable spheres. $S^7$ carries a smooth multiplication, making it the largest sphere with a group-like structure.
The 240 unit integer octonions are the 240 roots of $E_8$, split by scalar part as 2 + 126 + 112 into $\mathfrak{su}(2)$, $\mathfrak{e}_7$, and the bridging roots of $\mathfrak{e}_8$.