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de Bruijn’s pentagrids (2)
Last time we’ve seen that de Bruijn’s pentagrids determined the vertices of Penrose’s P3-aperiodic tilings. These vertices can also be obtained by projecting a window of the standard hypercubic lattice $\mathbb{Z}^5$ by the cut-and-project-method. We’ll bring in representation theory by forcing this projection to be compatible with a $D_5$-subgroup of the symmetries of $\mathbb{Z}^5$, which…
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Kasha-eating dragons
This semester I’m teaching a first course in representation theory. On campus, IRL! It’s a bit strange, using a big lecture room for a handful of students, everyone wearing masks, keeping distances, etc. So far, this is their only course on campus, so it has primarily a social function. The breaks in between are infinitely…
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de Bruijn’s pentagrids
In a Rhombic tiling (aka a Penrose P3 tiling) we can identify five ribbons. Opposite sides of a rhomb are parallel. We may form a ribbon by attaching rhombs along opposite sides. There are five directions taken by sides, so there are five families of ribbons that do not intersect, determined by the side directions.…
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this one’s for M.
and for everyone else feeling a bit lost in these tiring times. Photo credit: @itoldmythe
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Penrose’s aperiodic tilings
Around 1975 Sir Roger Penrose discovered his aperiodic P2 tilings of the plane, using only two puzzle pieces: Kites (K) and Darts (D) The inner angles of these pieces are all multiples of $36^o = \tfrac{180^o}{5}$, the short edges have length $1$, and the long edges have length $\tau = \tfrac{1+\sqrt{5}}{2}$, the golden ratio. These…