# Chapter ᚗ(V): The top landing *Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#ch-T24, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.* [Guide for AI readers](index.md) · [Its glossary cards](ch24-cards.md) · [Its references](ch24-refs.md) · [Previous chapter](ch23.md) · [Next chapter](ch25.md) *A footnote such as [r-24.3] is reference entry 24.3 and links to it. Section 24 of 28; chapters are named by their Ogham letter, never by number.* ᚗ – [Uilleann](ch24-cards.md#g-og-uilleann): associated with the honeysuckle, Section 24 ## Chapter ᚗ(V): The top landing > “First we lose the fact that every element is its own conjugate, then we lose commutativity, then we lose associativity, and finally we lose the division algebra property.” > — John C. Baez, on running the Cayley–Dickson construction from ℝ to ℂ to ℍ to 𝕆 to the sedenions · The Octonions, *Bulletin of the American Mathematical Society*, 2002 [r-24.14](ch24-refs.md#r-24.14) Formula: ([(e₃+e₁₀)(e₆−e₁₅)](ch24-cards.md#g-zd) + ([(aa)a − a(aa), a = e₃ + e₁₀](ch24-cards.md#g-powassoc)))/([‖e₃+e₁₀‖·‖e₆−e₁₅‖](ch24-cards.md#g-zdnorm)·([84](ch24-cards.md#g-zd84))) Key concepts: [Sedenions](ch24-cards.md#g-sedenions): sixteen real components, zero divisors, and powers that stay unambiguous. Zero here is a new kind of nothing: two [nonzero numbers whose product is zero](ch24-cards.md#g-zd). Finally, the top! My legs are burning. The airlock door is right beside us, and through the railings you can see how far we have climbed. Before we step through, take a look at this pair of nonzero factors whose product is zero. We have reached the sixteen-component [sedenions](ch24-cards.md#g-sedenions). A familiar cancellation argument will fail, and the Euclidean norm will no longer respect multiplication. Still, the construction has definite rules and surviving structure. Let’s see what this last doubling has changed. Field notes Sixteen real components, built under the book’s [Cayley–Dickson convention](ch24-cards.md#g-cdrule). The numerator shows a loss and a survival; the denominator measures the loss, once in the norm and once in a count. #### Numerator [(e₃+e₁₀)(e₆−e₁₅)](ch24-cards.md#g-zd) = 0; [zero divisors](ch10-cards.md#g-zerodiv), the headline loss of this landing and impossible in ℝ, ℂ, ℍ, or 𝕆. Take a = e₃ + e₁₀ and b = e₆ − e₁₅. Each has two nonzero coordinates, so neither is zero. Multiplying out gives four unit products, and they cancel in pairs: ab = e₃e₆ − e₃e₁₅ + e₁₀e₆ − e₁₀e₁₅ = e₅ − e₁₂ + e₁₂ − e₅ = 0. [r-24.2](ch24-refs.md#r-24.2) [r-24.3](ch24-refs.md#r-24.3) In ordinary numbers, ab = 0 with a ≠ 0 forces b = 0: multiply ab = 0 by the inverse of a and regroup, 0 = a⁻¹(ab) = (a⁻¹a)b = b. Our a does have an inverse, −a/2, since a² = −2. What fails is the regrouping. Because a⁻¹ is just a multiple of a, that regrouping is the alternative law the octonions kept on the previous landing, and the sedenions have lost it: here a⁻¹(ab) = 0, while (a⁻¹a)b = b. [r-24.2](ch24-refs.md#r-24.2) [r-24.3](ch24-refs.md#r-24.3) [r-24.6](ch24-refs.md#r-24.6) [(aa)a − a(aa), a = e₃ + e₁₀](ch24-cards.md#g-powassoc) = 0; [power associativity](ch24-cards.md#g-powassoc), a law that survives the climb. A weaker promise survives the loss of the alternative law: a single element still associates with itself, so every power of it means one thing however the brackets fall. The displayed cube is one instance: for our a, a² = −2, so (aa)a and a(aa) both come to −2a. The [sedenions](ch24-cards.md#g-sedenions) are also flexible, (ab)a = a(ba): when the same element stands at both ends of a product, the brackets still do not matter. [r-24.6](ch24-refs.md#r-24.6) [r-24.7](ch24-refs.md#r-24.7) #### Denominator [‖e₃+e₁₀‖·‖e₆−e₁₅‖](ch24-cards.md#g-zdnorm) = 2; the norms of the two zero divisors, chosen to show the norm is no longer multiplicative. Each factor has Euclidean norm [√2](ch04-cards.md#g-sqrt2), so the product of norms is 2, yet ab has norm 0. Thus ‖ab‖ ≠ ‖a‖‖b‖. The sixteen coordinates still have an ordinary positive norm; what fails is its compatibility with multiplication, as [Hurwitz’s theorem](ch23-cards.md#g-hurwitz) of 1898 said it must once the ladder climbed past eight. [r-24.4](ch24-refs.md#r-24.4) [r-24.5](ch24-refs.md#r-24.5) [84](ch24-cards.md#g-zd84) = 84; a count of zero-divisor patterns like the one above. Try sets of four distinct imaginary basis units and ways to pair them into two factors like those above. Forty-two sets work, each in exactly two pairings: 42 × 2 = 84. This counts patterns in the chosen basis, ignoring signs and factor order, not all [sedenion zero divisors](ch10-cards.md#g-zerodiv). [The glossary](ch24-cards.md#g-zd84) gives the full count, including the signs. [r-24.8](ch24-refs.md#r-24.8) [r-24.12](ch24-refs.md#r-24.12) The only fraction is (0 + 0)/168: 2 × 84 = 168, an ordinary nonzero integer. The same 168 turns up once more as the order of the symmetry group of [the Fano plane](ch23-cards.md#g-fano) that drew the previous landing’s multiplication table, though [the octonions](ch23-cards.md#g-octonions) of that plane have no [zero divisors](ch24-cards.md#g-zd) of their own. [r-23.10](ch23-refs.md#r-23.10) [r-24.2](ch24-refs.md#r-24.2) [r-24.13](ch24-refs.md#r-24.13) **Fine print.** Unlike the two previous landings, this denominator is a real integer, so no choice of left or right inverse is needed. The 84 counts cataloged patterns of basis units, not all [sedenion zero divisors](ch24-cards.md#g-zd), which form a much larger collection. **Looking down the tower.** Each doubling changed a different law. Complex numbers give up the order of [the reals](ch01-cards.md#g-reals), since no way of ranking them is compatible with their arithmetic (Baez counts a different first loss on the same step: a number stops being its own conjugate). [r-21.8](ch21-refs.md#r-21.8) [r-24.14](ch24-refs.md#r-24.14) [Quaternions](ch22-cards.md#g-quaternions) lose commutativity. [Octonions](ch23-cards.md#g-octonions) lose associativity. [Sedenions](ch24-cards.md#g-sedenions) lose alternativity and admit [zero divisors](ch24-cards.md#g-zd), and the multiplicative norm goes with them. Each system still has definite operations and worthwhile questions. The doubling recipe continues through 32, 64, and more components, beyond our chosen stopping point. What this chapter adds to the formula: (0 + 0) / 168 = 0 Just inside the airlock, the crew hands each of us an empty flight bag: “For anything you want to bring home.” Sit down and strap in; my legs have done enough climbing for today. Before any launch comes the pre-flight check, and I want to walk you through the controls. One of the gauges puts 27 nearer to zero than 3, so we had better find out what it means by “near.” Watch and read - [From Hamilton’s Quaternions to Graves & Cayley’s Octonions – Louis Kauffman](https://www.youtube.com/watch?v=pdidCZWgTYU) Gathering 4 Gardner talk: the whole doubling ladder told as a story, up to the rung where division fails - [Why Don’t Numbers Exist in 3D or 5D? The Surprising Truth About Dimensions](https://www.youtube.com/watch?v=yVMhI_6m6wI) Math Travel: why the well-behaved systems come only in dimensions 1, 2, 4, and 8, and what changes at 16 - [Sedenion](https://en.wikipedia.org/wiki/Sedenion) Wikipedia - [Cayley–Dickson construction](https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction) Wikipedia