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Kristopher Hite
1,611 followers -
PhD biochemist, blogger, skeptic, lover of green things
PhD biochemist, blogger, skeptic, lover of green things

1,611 followers
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What is in mind is a sort of Chautauqua...
"What is in mind is a sort of Chautauqua...that's the only name I can think of for it...like the traveling tent-show Chautauquas that used to move across America, this America, the one that we are now in, an old-time series of popular talks intended to edif...

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Hasan Minhaj at the White House Correspondents Dinner

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Fighting the Good Fight
I recall my father undergoing a major change in attitude about climate change 10 years ago after he attended a screening of Al Gore's An inconvenient Truth at the Chautauqua institution. The data combined with Al Gore's persuasive argument led my Dad to giv...

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Tesla taking orders for solar shingles April 2017
Since the moment Elon Musk announced TESLA would be offering solar-power generating roof shingles for houses last fall I have had a "solar shingles" Google-alert set up to notify me when news about this comes online. This morning I got an e-mail with a link...

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Desargues’ concertina

Suppose you draw two regular polygons with the same number of vertices, sharing the same centre, one polygon larger than the other. Allow for the possibility that one or both of these polygons are star polygons, where you draw edges not between consecutive vertices, but after skipping some fixed number of points.

The graph you get by joining up the vertices of the inner and outer polygons is known as an “I-graph”, and if the outer polygon is a normal polygon, it is known as a “Generalised Petersen graph”.

In 2012, three mathematicians proved that almost every I-graph, and every generalised Petersen graph, is a unit-distance graph: you can find a way to draw the graph so that all the edges have distance 1.

Žitnik, Arjana; Horvat, Boris; Pisanski, Tomaž, "All generalized Petersen graphs are unit-distance graphs", J. Korean Math. Soc. 49 (2012), No. 3, pp. 475–491

http://basilo.kaist.ac.kr/mathnet/thesis_file/JKMS-49-3-475-491.pdf

One example of a generalised Petersen graph is known as the Desargues graph. Here, the outer polygon is a decagon, while the inner polygon is a 10-pointed star where each vertex is joined to the one you get by adding 3, in a counter-clockwise numbering of the vertices. This turns out to be very easy to draw as a unit-distance graph: you just make the radii of the inner and outer polygons equal to the small and large golden ratios, which differ by 1. In most other examples, you need to introduce a twist between the rings of vertices, but here that isn't necessary.

You can read much more about the Desargues graph in this page by John Baez (which describes its construction in terms of relationships between subsets of a set of 5 elements):

http://math.ucr.edu/home/baez/networks/networks_14.html

The unit-distance version of the Desargues graph described above is not rigid, and in fact there is an 8-parameter family (up to overall rotations and translations) of ways of drawing the graph while maintaining the same edge lengths. The image below shows one highly symmetrical, 1-parameter family of unit-distance versions of the graph.

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Stephen Colbert to Stephen Miller "What the F** are you talking about?)*

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Senator Sheldon Whitehouse
<iframe width="595" height="335" src="https://www.youtube.com/embed/P1Bu29wR40o" frameborder="0" allowfullscreen></iframe>

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