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 exclusive-or 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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Banach-Tarski 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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Fast-growing functions revisited

There have been many exciting results proved by members of the Googology wiki, a website concerned with fast-growing functions. Some of the highlights include: Wythagoras’s construction of an 18-state Turing machine which takes more than Graham’s number of steps to … Continue reading

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4-input 2-output Boolean circuits

In 2005, Donald Knuth determined the minimum cost required to implement each of the 2^32 different 5-input 1-output Boolean functions as a circuit composed entirely of: 2-input gates (there are 16 of these), each of which has cost 1; 1-input … Continue reading

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That group of order 348364800

In nested lattices, we talked about the E8 lattice and its order-696729600 group of origin-preserving symmetries. In minimalistic quantum computation, we saw that this group of 8-by-8 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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