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 The exceptional Jordan algebra  Complex Projective 4Space on Pappian and Desarguesian planes
 Una Construcción Nueva: Definición – Mengenlehre on Von Neumann universe
 jasonhise64 on The BarnesWall lattices
 The BarnesWall lattices  Complex Projective 4Space on Minimalistic quantum computation
 The BarnesWall lattices  Complex Projective 4Space on Subsumptions of regular polytopes
Category Archives: Uncategorized
More quantum gates and lattices
The previous post ended with unanswered questions about describing the Conway group, Co0, in terms of quantum gates with dyadic rational coefficients. It turned out to be easier than expected, although the construction is much more complicated than the counterpart … Continue reading
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Minimalistic quantum computation
In the usual ‘circuit model’ of quantum computation, we have a fixed number of qubits, {q1, q2, …, qn}, and allow quantum gates to act on these qubits. The diagram below shows a Toffoli gate on the left, and an … Continue reading
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Nested lattices
1, 240, 2160, 6720, 17520, 30240, 60480, 82560, 140400, … These terms count the number of points at distance from the origin in the E8 lattice, a highly symmetric arrangement of points which Maryna Viazovska recently (in 2016) proved is … Continue reading
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Selfreplicator caught on video
In a previous article, an announcement was made of a complex selfreplicating machine (known as the 0E0P metacell) in a simple 2state cellular automaton. In the interim between then and now, Thomas Cabaret has prepared a most illuminating video* explaining … Continue reading
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(W^2 + X^2) / (W^2 + X^2 + Y^2 + Z^2)
The BoxMüller transform is a method of transforming pairs of independent uniform distributions to pairs of independent standard Gaussians. Specifically, if U and V are independent uniform [0, 1], then define the following: ρ = sqrt(–2 log(U)) θ = 2π V … Continue reading
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Complexity of integer multiplication almost solved
Whilst not quite as close as the proofs of the ternary Goldbach conjecture and bounded gaps between primes, there has been a quick succession of two important and somewhat complementary breakthroughs on the computational complexity of integer multiplication: Afshani, Freksen, … Continue reading
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Fully selfdirected replication
A new form of artificial life has been born — and there are no doubts that it directs its own selfreplication: So, what exactly is happening? At 0:06, the organism begins to sequentially construct four identical copies of itself. At … Continue reading
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6colourings of subsets of the plane
There has been further recent activity on the Chromatic Number of the Plane problem, with an eleventh research thread being spawned. Philip Gibbs has been able to 6colour a large disc (with diameter slightly greater than 4), and Aubrey de … Continue reading
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Atiyah’s problem
At the Heidelberg Laureate Forum three years ago, I took lots of selfies with Fields medallists, Abel prizewinners and Turing laureates. This included having a dinner in a castle with Leonard Adleman, pioneer of asymmetric cryptography: …and Endre Szemeredi of … Continue reading
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Sorting networks
Important note: whenever log is mentioned in this particular post, it is referring to the ceiling of the base2 (binary) logarithm. (Elsewhere on cp4space, when there isn’t this disclaimer, it refers to the basee (natural) logarithm.) For reasons that shall soon become … Continue reading
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