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 The exceptional Jordan algebra  Complex Projective 4Space on Pappian and Desarguesian planes
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Author Archives: apgoucher
4input 2output Boolean circuits
In 2005, Donald Knuth determined the minimum cost required to implement each of the 2^32 different 5input 1output Boolean functions as a circuit composed entirely of: 2input gates (there are 16 of these), each of which has cost 1; 1input … Continue reading
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That group of order 348364800
In nested lattices, we talked about the E8 lattice and its order696729600 group of originpreserving symmetries. In minimalistic quantum computation, we saw that this group of 8by8 real orthogonal matrices is generated by a set of matrices which are easily … Continue reading
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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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Fiveinput Boolean circuits
Over the past few weeks, I’ve been investigating Boolean optimisation. That is to say, given some circuit of logic gates that implements a particular ninput moutput function, find a more efficient circuit that implements the same function. In practical applications, … Continue reading
Posted in Boolean optimisation
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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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