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Monthly Archives: November 2013
Brouwer’s fixedpoint theorem
Many theorems in analysis have combinatorial analogues, which turn out to be equivalent (in the sense that one can be derived from the other, and viceversa, using significantly less maths than is necessary to prove either from first principles). A particularly … Continue reading
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Crash course in Gaussian integers
Yesterday, I attended the UKMT mentoring conference in Murray Edwards College (affectionately abbreviated to ‘Medwards’), which mainly consisted of an excellent dinner in the Fellows’ dining room and informal discussion about various topics. Speaking of mentoring, I recently prepared for one of … Continue reading
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Bounded gaps update
A few months ago, Yitang Zhang announced that there are infinitely many pairs of primes, separated by a distance no more than 70000000. This initiated an extensive collaborative effort (known as polymath8) to reduce this bound by optimising different parts of the proof, until … Continue reading
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Hearing the shape of a drum
(There appears to be a recent weeklong awkward silence on cp4space, partially due to the end of Season II of ciphers. Here’s an attempt to rectify it.) Suppose we have a square drum, consisting of a membrane fixed at its perimeter. The membrane … Continue reading
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Quantum logic gates
You are probably aware of classical logic gates, such as the Boolean AND and OR gates. The input(s) and output(s) of a logic gate take values in the set {0, 1}, usually identified with ‘false’ and ‘true’, respectively. One example … Continue reading
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Yf and only Yf?
There are plenty of triangle centres. One of the lesserknown triangle centres is Kimberling’s X(174), the Yff centre of congruence. It can be defined by considering lines known as isoscelisers, which are lines perpendicular to the angular bisectors of the triangle. If … Continue reading
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Gosper’s Pyrominia animation
As mentioned in a previous post, a pyritohedron is a (not necessarily regular) dodecahedron with pyritohedral symmetry. Up to scaling, any pyritohedron can be parametrised with the following coordinates: (±1, ±1, ±1); (±x, ±y, 0) and permutations thereof. For the … Continue reading
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