- Book embedding on Wikipedia (G+)
- The fractal geometry of plants (G+)
- A great combination of a success story of an individual woman in tech, a debunking of some nasty misogynist myths, and a collection of helpful resource links for others on the same path (G+)
- Trailer for a documentary on the shifting technology of graphic design from hand layout to computers (G+)
- Complaints about problems with peer review turn out to be far from new (G+)
- Curve-shortening flow on Wikipedia (G+)
- The Open Access Interviews: Timothy Gowers (G+)
- CMU’s computer science dean on its poaching problem (G+)
- From tilted decks of playing cards to hyperbolic geometry (G+)
- Springer journal copyrights are compatible with arXiv preprints (G+)
- Pinwheel zoom (G+)

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The rest of my photos include several from the conference excursion and dinner in the Valle de Guadalupe, Mexico's wine region.

- Ben Schneiderman's treemap art (G+)
- Eva Hild, ceramic artist whose works look like minimal surfaces (G+)
- Symmetric unfolding of the polytope formed by removing one vertex from a 4-cube (G+)
- Auto-resizing origami flowerpot (G+)
- Is it moral to zero-rate Wikipedia in third-world countries? (G+)
- Some tips to encourage more egalitarian contributions in discussion-based classes (G+)
- Linear probing, now a good article on Wikipedia (G+)
- Hexagonal structures in nature (G+)
- Portrait of the scientist as an infant (shadowplay GIF; G+)
- Fake university exposes visa fraud (G+; discussion involves the "true" purpose of degree programs)
- Do open peer-review systems play into sexism? (G+; and if you think they might, be sure to contribute to the arXiv user survey before it closes on April 26)
- Andrew Gleason, a mathematician who made contributions to Lie group theory, quantum mechanics, Ramsey theory, coding theory, cryptanalysis, and calculus reform, among others (G+)
- Teen hackers in the early 1980s (G+)
- The degree-6 surface with the maximum possible number of nodes (65 of them). But for higher degrees the maximum number is still unknown (G+)
- 3d model of an imagined city, made entirely out of paper by Katsumi Hayakawa (G+)

- John Oliver on the reopening of the cryptowars (G+)
- Bad ad-screening hygiene exposes thousands to malware so you should use an ad-blocker to be safe (G+)
- Turkey equates petition-signing to terrorism, jails three professors, hounds others from their positions (G+)
- Printable magnets (G+)
- The distribution of cell shapes in Voronoi diagrams of random points in space (G+)
- The link between engineering education and terrorism (G+)
- Sphere packing solved in dimensions 8 and 24 (G+)
- Split decomposition of graphs (G+)
- Wrinkles and crumples make graphene better (G+)
- Integer sorting (G+)
- Visual proof of the arithmetic-geometric mean inequality (G+)
- Lego Escher (G+)
- Nalini Joshi on gender discrimination in Australian mathematics (G+)
- Addictive abstract puzzle game based on the theory of Thompson groups (G+)

I recently asked my data structures class the following question: suppose you fill in an *n*-cell array by a random permutation. What is the probability that the result is a valid binary min-heap?

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- Why aiming to run a university like a business is a worse idea than it sounds like (G+)
*Discrete Analysis*journal publishes its inaugural issue (G+)- Australian trade controls threaten academic freedom (G+)
- Interview with Michael Atiyah with some nice insights into collaborative mathematical research (G+)
- The Frankl–Rödl graph and the challenge it poses to the Unique Games Conjecture (G+)
- Discovery of the Copperhead, a new small c/10 spaceship in Conway's Life (G+)
- Crossword plagiarism (G+)
- Academic journal forces pseudonymous authors to reveal their names (G+)
- Möbius transformation of a Doyle spiral circle packing (G+)
- David Johnson (1945–2016) (G+)
- Using topology to simplify a real-world sewing problem (G+)
- Yet another academic sexual harrassment case brings down the Berkeley law dean (G+)
- There is no cloud. It's just someone else's computer. (G+)
- Nonrandomness in the final digits of consecutive primes (G+)
- A fancier spirograph and its online simulator (G+)

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- SoCG accepted paper titles (G+)
- University of California President Janet Napolitano spies on all campus emails and other internet traffic (G+)
- Sheep flocking patterns (G+)
- Using the Thue–Morse sequence to pour coffee (G+)
- FUN and COCOON submission deadlines extended to early March (G+)
- Frozen soap bubbles (G+)
- Group theorist Dimitri Leemans returns to Europe after NZ denies residency to his son (G+)
- Why what the FBI claims to want Apple to do is impossible (G+)
- Lio blows geometric soap bubbles (G+)
- Mathematical jokes: theorems and proofs that don't look like they should be valid, but are (G+)
- Matchstick puzzles (G+)

*x*is a vertex that can be reached by a walk from

*x*. However, it's also possible to turn this around, and define trees in terms of their ancestor-descendant relations, with the parent-child relation derived from that. In the turned-around definition, a tree is a partial order with a designated root element that is a predecessor of every element, such that the predecessors of each element are totally ordered. The parent of an element would then be the maximum of its predecessors. So graph-theoretic finite trees and order-theoretic finite trees are really the same things as each other, in a different disguise. But that's no longer true when we go from finite to infinite: we can define trees in either of these two ways, but we get different things.

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**( More photos of Black Star Canyon )**

- Numberphile video on triangle centers (G+)
- New US visa requirements for travelers from visa-waiver countries who have visited Iran, may especially effect computational geometers who have been to the Winter School on Computational Geometry (G+)
- Lecture notes from de Bruijn's course on combinatorics have an unlikely host (G+)
- Undecidability of an energy gap in arrays of quantum devices (G+)
- Analog rather than digital production of chaotic/fractal images, using video feedback (G+)
- New Wikipedia book on perfect graphs (G+)
- Congratulations to the computer scientists newly inducted into the NAE (G+)
- 3d-printing a digital sundial gnomon (G+)
- Zombies vs humans, visualization of a continuous pursuit-evasion model (G+)
- Rectangle packing on the surface of an integer cube (G+)
- List of accepted papers to STOC 2016 (G+)
- The Kansas and Louisiana public university systems race to the bottom. But maybe the threat to the football team can act as a parachute? (G+)
- Reversing the curve-shortening flow can send nice curves into very messy singularities (G+)
- Ed Pegg's new tetrahedrally-symmetric surface of constant width (G+)

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- Cairo tiling Legos (G+)
- Zooming in spherical videos using Möbius transformations, with bonus spherical Droste effect (G+)
- Corporate-sponsored chocolate-milk press-release-research at the University of Maryland (G+)
- The power of two choices works very well for load balancing with stale data (G+)
- Harvard's library of colors (G+)
- 3d color printing with layers of paper (G+)
- What is the name of the trillium theorem? (G+)
- Women outnumber men in USC's game design program (G+)
- New York Times obituary for Marvin Minsky (G+)
- Java applets are dead. So what do we use as a replacement for Cinderella? (G+)
- Geometric mean of ranks aggregation in sports competition scoring: unfair because it disobeys independence of irrelevant alternatives? (G+)
- Physics arXiv crank filter had two false positives. Should we do the same in CS? (G+)
- What started out as a boring spat between two mathematical physicists has turned into an interesting discussion of the role of blogs and other less-formal writings in scientific research and publishing (G+)

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*d*-dimensional convex polytopes

*P*, the covering graph of its face lattice is the graph of a (

*d*+ 1)-dimensional convex polytope, which for lack of a better name I'll call the face incidence polytope, or incidence polytope for short (although that shorter phrase does already have a different meaning).

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The binary strings of a given length, like the length-8 string "11011110" in the name of my blog, can be thought of as naming the vertices of a hypercube of the same dimension: each bit is one of the Cartesian coordinates of a vertex. In the same way, binary strings with wildcard characters, like "11***1*0", can be thought of as naming the nonempty faces of the hypercube; the number of stars gives the dimension of the face, up to the string "********" which represents the whole cube. But there's one more face, the empty set Ø, which cannot be represented in the same way.

As with the collection of faces of any polyhedron, the faces of a hypercube can be partially ordered by inclusion, and this partial order forms a lattice: every family of faces has a unique meet (its greatest lower bound, the intersection of all the faces), and a unique join (its least upper bound, the unique minimal face that contains all of them). For instance, the meet of two opposite sides of an ordinary 3-dimensional cube (for instance the two sides **0 and **1) is the empty set (that's why Ø needs to be a face) and the join of the same two opposite sides is the whole cube ***. This is the face lattice of the hypercube. (The hypercube itself can also be viewed as a face lattice of another kind of polyhedron, a simplex.)

Here's an example, for the face lattice of a square (a 2-dimensional cube). The inclusion ordering is shown by the edges, and each lattice element is labeled both by the part of the square it represents and by the corresponding wildcard string.

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