Richard Green

I'm a professor of mathematics at the University of Colorado Boulder.
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[Richard Green @edfromo]

How many cylinders can touch each other?

What is the largest number of identical infinite cylinders that can be arranged in three dimensional space so that every two of them touch? It is at least 7, and at most 10.

https://t.co/Oqfv0z4Pvy

#math #maths #apieceofthepi #substack

[Richard Green @edfromo]

The golden ratio as a number base

The golden ratio, φ=(1+√5)/2, can be used as a number base. Integers that have a symmetric representation in base φ turn out to have some interesting properties.

https://t.co/yKB883w8QI

#apieceofthepi #math #maths

[Richard Green @edfromo]

Matroid bingo is a game of chance that can be used to define the mathematical concept of a matroid. The probabilities involved in the game have some surprising properties.

https://t.co/ARvRS6EshB

#apieceofthepi #math #maths #substack

[Richard Green @edfromo]

Spiral Sudoku

There is a unique way to fill this 5 by 5 grid with the digits 1 to 5 so that each digit appears once in each row and each column, and so that the digit k appears in exactly k of the circles.

https://t.co/fUu5j3a40l

#math #maths #substack

[Richard Green @edfromo]

Games in projective space

The design of the game Dobble (or Spot It!) is based on the structure of a finite projective plane. This is a discrete version of two dimensional projective space, in which parallel lines do not exist.

https://t.co/SRQPvwoyxj

#math #maths #substack

[Richard Green @edfromo]

Intersections of chords of a circle

Given two random chords of a circle, what is the probability that their intersection lies within distance r of the centre? What do we even mean by a “random” chord?

https://t.co/0F8dsd9oF4

#apieceofthepi #substack #maths #math #probability

[Richard Green @edfromo]

Geodesics and polyhedra

A geodesic on a cube is a curve on the cube that becomes a straight line when the cube is folded flat. There are three types of geodesics on a cube, and 15 types of “quasigeodesics”.

https://t.co/2JQQJRlgP1

#apieceofthepi #substack #geometryy #maths

[Richard Green @edfromo]

Venn diagrams and Winkler’s conjecture

Can a simple Venn diagram on n sets always be extended to a simple Venn diagram on n+1 sets by adding a suitable curve? Apparently not!

https://t.co/BfMCziAWUt

#apieceofthepi #math #maths #substack

[Richard Green @edfromo]

@mk270 @onehappyfellow That is the best description I’ve seen of mathematicians’ use of “morally”. I don’t know where it came from, and it is very odd usage. What does it say about mathematicians’ morals, I wonder?

[Richard Green @edfromo]

Which way is the bike going?

Can you deduce the direction of travel of a bike by the shape of the tracks it leaves? Sherlock Holmes thought so, but the answer is somewhat subtle.

https://t.co/EboaBw4pbE

#apieceofhtepi #substack #math #maths

[Richard Green @edfromo]

Random coprime numbers

The probability that two random large integers have no factors in common is 6 divided by π squared, which is about 60.8%.

https://t.co/VLksqIVxtc

#apieceofthepi #piday #math #maths #substack

[Richard Green @edfromo]

Rubik’s abstract polytopes

If we make a 3×3×3 face-turning Rubik’s cube in the shape of a different Platonic solid, how many symmetries does it have? How about if we do it in higher dimensions?

https://t.co/OlCKI9FlaO

#apieceofthepi #maths #math #substack

[Richard Green @edfromo]

Counting dimer tilings

How many ways can you cover a square grid with non-overlapping 2 by 1 dominoes, leaving at most one square empty?

https://t.co/5hQeYyHZkE

#apieceofthepi #math #maths #substack

[Quanta Magazine @QuantaMagazine]

Since the 1960s, mathematicians have been trying to figure out the biggest shape that can fit through an L-shaped hallway. A recent proof uncovered the answer without any computer assistance. It instead took a fresh approach to solving optimization problems.
https://t.co/hF9HXJ0g4X

[Quanta Magazine @QuantaMagazine]

The next time you move, consider mathematician Joseph Gerver’s 18-piece sofa. It’s the biggest shape that can slide down an L-shaped hallway… though maybe not the easiest to assemble. https://t.co/i3HXI0iTmo

[Quanta Magazine @QuantaMagazine]

If you’ve ever moved into a new home, then you know how difficult it can be to steer bulky furniture through narrow hallways or around awkward corners. A new proof reveals the biggest shape that can slide down an L-shaped hallway. Richard Green reports: https://t.co/i3HXI0iTmo

[Richard Green @edfromo]

The moving sofa problem

Jineon Baek recently announced a proof that Gerver’s sofa is the largest shape that can be slid around a right-angled hallway of unit width. @QuantaMagazine has just published an article by me about this.

[Quanta Magazine @QuantaMagazine]

If you’ve ever moved into a new home, then you know how difficult it can be to steer bulky furniture through narrow hallways or around awkward corners. A new proof reveals the biggest shape that can slide down an L-shaped hallway. Richard Green reports: https://t.co/i3HXI0iTmo

[Richard Green @edfromo]

Identifying bottlenecks in networks

A complicated network often consists of highly connected regions that are linked to each other by bottlenecks. How can we identify bottlenecks computationally?

https://t.co/TEX7cNioaP

#substack #math #maths

[Richard Green @edfromo]

The mathematics of the game Waffle

Waffle is a word puzzle game along the lines of Wordle. In order to solve a game efficiently, it helps to know something about abstract algebra and the symmetric group.

https://t.co/uFkY950PzD

#substack #maths #math

https://t.co/uFkY950PzD

[Richard Green @edfromo]

On @substack Turning a triangle into a square

There is a well-known way to cut an equilateral triangle into four polygonal pieces and slide the pieces back together to form a square. It is impossible to do this with three pieces.

https://t.co/njz8M31Cnj

#math #substack