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I am the founder of Hatsya Limited, which provides mathematical and technological solutions to realworld problems.

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Author Archives: apgoucher
Associative universal gates
The Boolean function NAND is famously universal, in that any Boolean function on n inputs and m outputs can be implemented as a circuit composed entirely of NAND gates. For example, the exclusiveor operation, A XOR B, can be written … Continue reading
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Another two rational dodecahedra
Since finding one rational dodecahedron inscribed in the unit sphere, I decided to port the search program to CUDA so that it can run on a GPU and thereby search a larger space in a reasonable amount of time. Firstly, … Continue reading
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BanachTarski and the Axiom of Choice
Tomasz Kania and I recently coauthored a paper about Banach spaces. The paper makes extensive use of the axiom of choice, involving a transfinite induction in the proof of Theorem B as well as several appeals to the fact that … Continue reading
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Rational dodecahedron inscribed in unit sphere
Moritz Firsching asked in 2016 whether there exists a dodecahedron, combinatorially equivalent to a regular dodecahedron, with rational vertices lying on the unit sphere. The difficulty arises from the combination of three constraints: The twelve pentagonal faces must all be … Continue reading
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Fastgrowing functions revisited
There have been many exciting results proved by members of the Googology wiki, a website concerned with fastgrowing functions. Some of the highlights include: Wythagoras’s construction of an 18state Turing machine which takes more than Graham’s number of steps to … Continue reading
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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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