A Piece of the Pi: mathematics explained

A Piece of the Pi: mathematics explained is a Substack aimed at explaining complex mathematical concepts to intelligent general readers. It covers a wide range of mathematical topics such as number theory, geometry, probability, and algebra, translating intricate mathematical phenomena and theories into accessible insights.

Number Theory Geometry Algebra Probability Combinatorics Mathematical Puzzles Mathematical Physics Fractals and Patterns Game Theory Mathematical Biology

The hottest Substack posts of A Piece of the Pi: mathematics explained

And their main takeaways

The graphic nature of groups

42 implied HN points • 12 Jul 23
  1. Character theory in abstract algebra is related to group theory and studies symmetry.
  2. Character theory has applications in the study of molecular vi...
  3. This post is for paid subscribers on apieceofthepi.substack.com.

Misleading diagrams

30 implied HN points • 20 Oct 23
  1. The post discusses misleading diagrams in geometry
  2. The image shows a triangle divided into smaller shapes that can be rearranged to alter the area
  3. Access to the full post is limited to paid subscribers

Leaf functions of graphs

24 implied HN points • 16 Dec 23
  1. Trees in different lattice structures have leaves that touch only a limited number of other cells.
  2. The number of leaves a tree with a certain number of vertices can have is a focus of study in graph theory.
  3. Leaf functions have been computed for various lattice structures like square, hexagonal, triangular, and even cubic lattices.

The symmetry of musical key changes

42 implied HN points • 01 Jul 23
  1. The _Tonnetz_ diagram represents musical tonal space
  2. The diagram was developed by mathematician Leonhard Euler in 1739
  3. Paid subscribers have access to more detailed content

Ham sandwiches and necklace splitting

42 implied HN points • 28 Jun 23
  1. There are two points on Earth's surface that always have the same temperature and pressure.
  2. This phenomenon is a result of the Borsuk-Ulam theorem.
  3. This post is for paid subscribers only.
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The Fifteen Puzzle and its variants

36 implied HN points • 29 Jul 23
  1. The Fifteen Puzzle is believed to be the oldest type of sliding puzzle invented in the 1870s.
  2. It consists of fifteen squares that can be slid around a 4 by 4 grid.
  3. The post is for paid subscribers only, but you can subscribe to access it.

Rock, Paper, Scissors, Lizard, Spock

36 implied HN points • 19 Jun 23
  1. The well-known rules of Rock, Paper, Scissors
  2. People often use a mnemonic to remember the rules
  3. This post is for paid subscribers only

Locked polyomino tilings

30 implied HN points • 02 Aug 23
  1. The post discusses locked polyomino tilings based on a specific study by Jamie Tucker-Foltz.
  2. The image in the post shows the only known locked tiling of a square grid by Tetris pieces.
  3. The post is for paid subscribers only.

Winning the lottery with the Fano plane

30 implied HN points • 26 Jul 23
  1. The Fano plane concept can be used to win the lottery by selecting unique numbers.
  2. In the UK's National Lottery, players choose six numbers out of 1 to 59 for a chance to win.
  3. During the draw, six random balls are selected from a range of numbers, determining the winning combination.

Loaded dice and unfair polynomials

30 implied HN points • 19 Jul 23
  1. Throwing a total of 7 with two fair dice is six times more likely than throwing a total of 12.
  2. To roll a 12, both dice need to show a 6.
  3. The concept of loaded dice and unfair polynomials explained.

The Hat tiling

36 implied HN points • 16 May 23
  1. Aperiodic monotiles are shapes that can tile the plane without translational symmetry.
  2. The existence of such shapes was an open question that has been recently answered.
  3. This post is for paid subscribers only.

Numbers and their reversals

30 implied HN points • 05 Jul 23
  1. This post discusses reversing digits of numbers.
  2. G.H. Hardy noted unique examples of four-digit numbers.
  3. The post is for paid subscribers only.

Corridor numbers, Fibonacci numbers, and Pascal’s triangle

12 implied HN points • 25 Feb 24
🔬 Science Mathematics Geometry Chess Induction
  1. Corridor numbers count ways to take diagonal steps down a corridor with fixed width. The numbers in each box form Fibonacci numbers when summed vertically.
  2. Fibonacci sequence is generated by summing the previous two terms. In the context of corridor numbers, Fibonacci numbers represent different routes to specific boxes.
  3. Pascal's triangle has rows starting and ending with 1, where each entry is the sum of two nearest entries from the row above. Circular Pascal arrays relate to corridor numbers and can produce Fibonacci numbers when subtracting specific entries.

What is a set?

24 implied HN points • 19 Aug 23
  1. A set in mathematics is a collection of objects like fruits or prime numbers.
  2. The key property of a set is that each element is unique.
  3. The post discusses the concept of sets and is for paid subscribers.

The random graph

30 implied HN points • 19 May 23
  1. The term 'graph' in mathematics can have different meanings, but in combinatorics, it refers to a set of points connected by edges.
  2. The article discusses the concept of 'random graph' and its relevance in mathematics.
  3. This post is for paid subscribers only, highlighting the need for a subscription to access the content.

Token-moving puzzles

24 implied HN points • 15 Jul 23
  1. Token-moving puzzles are one-player games.
  2. In these puzzles, tokens are placed on a grid.
  3. Legal moves involve picking up and relocating tokens.

The best denominations for coins

24 implied HN points • 07 Jul 23
  1. The post discusses the best denominations for coins.
  2. It suggests how to minimize the number of coins used when giving change.
  3. Paid subscribers can access the full content.

The "Lights Out" game

24 implied HN points • 12 Jun 23
  1. The 'Lights Out' game is played on a 5 by 5 grid of lights that can be turned on or off.
  2. In the game, a move involves toggling the state of a selected light and its adjacent lights.
  3. This post is for paid subscribers only.

The moving sofa problem

24 implied HN points • 09 Jun 23
  1. The moving sofa problem is about finding the largest planar shape that can be moved around a right-angled corner of unit width.
  2. This problem was first posed by Leo Moser in 1966 and remains unsolved in geometry.
  3. The post about the moving sofa problem is available for paid subscribers only.

Regular polytopes

24 implied HN points • 05 Jun 23
  1. There are five regular solids in three dimensions: tetrahedron, octahedron, icosahedron, cube, and dodecahedron.
  2. The two-dimensional case is a topic discussed in the post but further details are only available to paid subscribers.
  3. The author provides a link to subscribe for exclusive content.

How to build a computer using origami

17 HN points • 20 Sep 23
  1. Origami can be used to build computers
  2. Flat origami involves folding paper in a way that the object lies flat
  3. Paid subscribers can access a detailed post on building a computer using origami

Bonus content: The best denominations for coins

18 implied HN points • 07 Jul 23
  1. The US dime coin can be confusing because its value is not easily distinguishable by appearance.
  2. This post is for paid subscribers only.
  3. The post discusses the best denominations for coins.

Fractal squares

18 implied HN points • 21 Jun 23
  1. The post discusses the construction of the Sierpiński carpet, a fractal by Wacław Sierpiński in 1916.
  2. The fractal is built by repeatedly removing squares from the previous iteration.
  3. The post is available only for paid subscribers, and there is an option to subscribe on the page.

Prime knots and knot mosaics

18 implied HN points • 17 Jun 23
  1. Knots in mathematics are like closed loops in three-dimensional space.
  2. Examples of knots can be more complex like trefoil knots.
  3. The post discusses the concept of prime knots and knot mosaics.

The Golay code

18 implied HN points • 14 Jun 23
  1. Errors are inevitable in information transmission.
  2. NASA used the Golay code to transmit high-resolution pictures of Jupiter and Saturn.
  3. The post is for paid subscribers only.

The nonlinear theory of sound

18 implied HN points • 26 May 23
  1. The post discusses the nonlinear theory of sound
  2. It explains how vibrations of a musical instrument create oscillations in air pressure
  3. The frequencies of these oscillations correspond to music

Dissecting polygons into squares and rectangles

12 implied HN points • 28 Sep 23
  1. The Wallace–Bolyai–Gerwien theorem states that polygons with the same area can be cut and rearranged.
  2. This post is for paid subscribers only.
  3. Geometry can involve dissecting polygons into squares and rectangles.

Misère X-only tic-tac-toe

18 implied HN points • 11 May 23
  1. In misère X-only tic-tac-toe, players aim to lose the game by playing as X against an opponent also playing as X.
  2. The concept of misère X-only tic-tac-toe introduces a unique challenge where winning means losing.
  3. This post discusses the intriguing idea of misère X-only tic-tac-toe and the strategy behind trying to lose the game.

Flexible frameworks

18 implied HN points • 09 May 23
  1. The post discusses the concept of flexible frameworks.
  2. It highlights a recent paper on flexibility and rigidity of triangular and parallelogram frameworks.
  3. The content seems to be exclusively for paid subscribers.

The horned sphere and the game of Hex

12 implied HN points • 24 Aug 23
  1. The Alexander horned sphere is an unconventional way to embed a sphere in 3D space.
  2. J.W. Alexander first described the horned sphere in 1924.
  3. The post discusses the relationship between the horned sphere and the game of Hex.

The mathematics of card tricks

12 implied HN points • 09 Aug 23
  1. The post discusses the mathematics behind card tricks.
  2. It talks about selecting a card, placing it on top of the deck, and cutting the deck.
  3. The content is for paid subscribers only.

Bonus content: The symmetry of musical key changes

12 implied HN points • 01 Jul 23
  1. The post discusses the symmetry of musical key changes.
  2. Euler and Hugo Riemann would have seen the Tonnetz as an infinite two-dimensional network.
  3. The content is available for paid subscribers only.

Amidakuji!

12 implied HN points • 07 Jun 23
  1. Amidakuji is a method of randomly pairing elements from two sets of the same size.
  2. Amidakuji is known in Korea as another name.
  3. The post about Amidakuji is only available for paid subscribers.

Emergent self-replicating structures

12 implied HN points • 02 Jun 23
  1. The article discusses emergent self-replicating structures.
  2. The main focus is on Byl's loop, a cellular automaton developed by John Byl.
  3. This post is for paid subscribers only.

The optimal dartboard

12 implied HN points • 24 May 23
  1. The post is about the concept of an optimal dartboard setup.
  2. The author references a question by comedian Steven Wright about the order of the alphabet.
  3. This post is intended for paid subscribers only.

Cutting a rectangle into squares

6 implied HN points • 31 Aug 23
  1. You can cover a rectangle with squares by using 1 by 1 squares when both sides are integers.
  2. The challenge is to cover the rectangle using squares without using any 1 by 1 squares.
  3. The content of the post is available to paid subscribers only.

Bonus content: The graphic nature of groups

6 implied HN points • 12 Jul 23
  1. The post discusses the graphic nature of groups
  2. The content is for paid subscribers only
  3. There are options to subscribe or sign in as a paid subscriber

Parking functions and the permutohedron

6 implied HN points • 29 May 23
  1. The post discusses parking functions and the permutohedron.
  2. It involves a scenario of cars trying to park in preferred spaces on a one-way street.
  3. This post is available for paid subscribers only.

No three in a line

3 HN points • 25 Nov 23
  1. The post discusses the concept of placing 20 points in a 10x10 grid so that no three points are in a line.
  2. The image in the post visually represents the placement of the points in a grid.
  3. The content is available for paid subscribers only.