Check it

The book has a house rule

Nothing goes into the book without a proof or an exhaustive check over the 512 landas. Here you can repeat those checks yourself: they run in your browser, with the book's library, without sending anything to any server.

How to read it

• «Exhaustive» means every case is tested. «Sample» means random cases are tested, always the same ones (fixed seed), and it does not replace the proof: it reinforces it.
• Associativity can be checked by brute force, over the 134,217,728 possible triples. It takes a few seconds, can be cancelled, and its result follows from the proof anyway: ∗ is an XOR.

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exhaustive

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Brute force (134,217,728 triples)

512 landas, 48 hamiltonians

≅ (ℤ₂)⁹

brute force

sample

50 alternative rests

sample + orthogonality

20 random functions

Structure of the landas

There are 512 distinct landas, 48 hamiltonians, each is a chain of neighbors, and the hemisphere changes exactly when the dimension repeats.

Total and single-valued

For all 262,144 pairs there is a single result, and it is one of the 512 landas.

Commutative

a ∗ b = b ∗ a for all 262,144 pairs.

With a neutral element

a ∗ rest = a for all 512 landas.

An involution

a ∗ a is the rest for all 512 landas.

Latin square (lossless)

For each a, the map b ↦ a ∗ b is a bijection: every row of the table contains the 512 landas once.

Isomorphism with (ℤ₂)⁹

c(a) = index(a) XOR index(rest) is a bijection onto the 9-bit numbers, and c(a ∗ b) = c(a) XOR c(b) for all 262,144 pairs.

Associative

(a ∗ b) ∗ c = a ∗ (b ∗ c). A sample of one million triples, or all 134,217,728 by brute force.

The book's definition

a ∗ b is a moved by the movement that takes the void to b, for all 262,144 pairs.

Invariant

If a and b are moved by the same movement, a ∗ b does not change. A sample of 500,000 cases.

The mirror

x ∗ mirror is x with each pole swapped for its dimensional opposite, for all 512 landas.

Another rest

With another landa e′ as rest, a ∗′ b = (a ∗ b) ∗ e′ is commutative, associative, an involution, and has e′ as its neutral element.

Fourier characters

χ_w(a ∗ b) = χ_w(a) · χ_w(b) (one million cases), χ_w(x) = χ_x(w) for all 262,144 pairs, and orthogonality of the characters.

Walsh–Hadamard transform

The fast transform in nine stages matches its definition, inverts exactly and satisfies Parseval.

Translation of the 9-bit hypercube

x ↦ x ∗ m preserves the number of bits in which two landas differ (Hamming distance), for any m. A sample of 500,000 cases.