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

105 trillion digits of pi

60 implied HN points • 15 Mar 24
🔬 Science Mathematics Algorithms
  1. The number pi has now been calculated to 105 trillion decimal places using the Chudnovsky algorithm over 75 days.
  2. Ramanujan's formula for pi has been expanded and improved upon over the years, with the Chudnovsky brothers developing a formula that computes pi to 13 decimal places.
  3. Bellard's formula and the BBP formula provide ways to compute specific digits of pi without having to calculate all earlier digits, making validations faster and more efficient.

Stacks of cards and the harmonic series

54 implied HN points • 28 Jan 24
  1. Correctly stacking objects relies on the center of mass being balanced.
  2. The size of overhang in a stack of objects follows a pattern related to the harmonic series.
  3. The harmonic series explains how an overhang in stacked objects can theoretically be infinitely large.

The McNugget monoid

60 implied HN points • 21 Jan 24
  1. The McNugget monoid is a set of numbers that can be made using additive combinations of 6, 9, and 20
  2. Numerical monoids, like the McNugget monoid, have specific properties, such as containing all but finitely many natural numbers
  3. The McNugget monoid can also be represented as a combination of 6, 9, and 20, with particular emphasis on these numbers' unique role in the set

Picturing symmetry

24 implied HN points • 04 Mar 24
🔬 Science Symmetry Mathematics Group Theory
  1. The order in which symmetries are applied can significantly affect the final result, as shown through reflections and rotations of a square.
  2. Using Cayley graphs can help visualize and calculate products of symmetries.
  3. In symmetry operations, combining reflections and rotations follows specific rules, similar to adding odd and even numbers. Grouping rotations and reflections can simplify understanding complex symmetries.

The infinite monkey theorem

18 implied HN points • 11 Mar 24
🔬 Science Mathematics Literature Research Experiment Evolution
  1. The infinite monkey theorem states that given enough time and randomness, a monkey could type out the complete works of Shakespeare on a keyboard.
  2. Generating longer phrases by random means, as shown in simulations, becomes exponentially more difficult as the phrase length increases.
  3. The famous infinite monkey paradox has been explored through history, including Cicero's speculation in 45 BC and modern computer simulations using actual monkeys with disappointing results.
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Edge colourings and Archimedean lattices

24 implied HN points • 18 Feb 24
🔬 Science Mathematics Graphs Symmetry
  1. Edge colorings of graphs are not just recreational, but have practical applications in quantum technology.
  2. Graphs can be colored either by edges or by vertices, with different requirements for each coloring approach.
  3. Vizing's Theorem states that a graph can be edge colored with either the maximum degree or the maximum degree plus one colors.

Pyritohedral symmetry

24 implied HN points • 11 Feb 24
🔬 Science Symmetry
  1. Pyritohedral symmetry is a unique type of symmetry displayed by certain crystal structures, like pyrite crystals, which have irregular pentagonal faces forming an irregular dodecahedron.
  2. Pyritohedral symmetry involves various rotations and axes of rotational symmetry, represented by colorful lines highlighting vertices and edges. This symmetry forms a group known as A_4.
  3. Complex crystals like the Holmium-Magnesium-Zinc quasicrystal exhibit even more intricate symmetries, such as those of a regular dodecahedron, represented by larger rotational symmetry groups like A_5.

Grids and folding curves

48 implied HN points • 31 Dec 23
  1. The grid in the picture has various shapes touching each point, such as a triangle, square, hexagon, and another square
  2. Grids can be understood through edge classes and prototiles, which define transitions between edges
  3. L-systems model biological growth and can result in self-avoiding curves and plane-filling patterns

Guillotine and diagonal rectangulations

36 implied HN points • 14 Jan 24
  1. The rectangulations shown are examples of guillotine and diagonal rectangulations.
  2. Studying rectangulations can reveal interesting patterns like left-right bricks and windmills.
  3. Counting types of rectangulations can involve understanding rational and algebraic generating functions.

Generalizations of Euler’s Tonnetz

24 implied HN points • 04 Feb 24
  1. The Tonnetz is a tessellation of triangles labeled by musical notes in a repeating pattern.
  2. Root systems like A2, B2, and G2 play a significant role in mathematics and physics.
  3. Duality in the root systems can switch major and minor keys in the Tonnetz.

Curves and L-systems

77 HN points • 03 Nov 23
  1. The image shows the Gosper curve discovered by Bill Gosper in 1973.
  2. The Gosper curve is a space-filling curve that converges to a specific shape.
  3. The article is for paid subscribers only.

Self-complementary ideals

36 implied HN points • 05 Jan 24
  1. Self-complementary ideals have symmetric properties when reflecting in planes and mirrors.
  2. The flip graph of self-complementary ideals shows how they can be transformed with specific rules.
  3. Connections between self-complementary ideals can be proven and depend on dimensions and odd-even numbers.

Diophantine m-tuples

84 implied HN points • 13 Oct 23
  1. Diophantine m-tuples involve sets of integers with a special property related to squares.
  2. A Diophantine quadruple is a set of four integers where multiplying any two and adding 1 results in a square number.
  3. The post on Diophantine m-tuples is available only to paid subscribers.

Derangements and the number e

60 implied HN points • 16 Nov 23
  1. The post discusses derangements and the number e.
  2. It presents a probability scenario with 20 students and homework assignments.
  3. Access to the full post is for paid subscribers only.

Fractional Sudoku

72 implied HN points • 27 Oct 23
  1. The post is about Fractional Sudoku.
  2. The Sudoku puzzle in the post is completed using colors instead of numbers.
  3. The solution is valid because each color appears exactly once in each row and column.

Sandpiles and the Vicsek fractal

36 implied HN points • 24 Dec 23
  1. The sandpile model involves stacking tokens on a grid and toppling them if the stack gets too high.
  2. Designating a sink vertex in the sandpile model on a graph reveals interesting properties.
  3. The Vicsek fractal graph shows a 1 in 4 probability of causing an endless avalanche when adding a token to a stable configuration.

Corridor numbers, Fibonacci numbers, and Pascal’s triangle

12 implied HN points • 25 Feb 24
🔬 Science Mathematics Geometry Chess
  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.

The combinatorics of redistricting plans

36 implied HN points • 07 Dec 23
  1. Tilings of grids in redistricting help visualize dividing voters into districts.
  2. Randomly picking redistricting plans is challenging due to vast possibilities and poor district shape.
  3. Using tests like the Polsby-Popper score helps evaluate district compactness in redistricting.

Benford’s Law

66 implied HN points • 07 Sep 23
  1. Benford's Law suggests that in many datasets, the leading digit is likely to be small, especially the digit 1.
  2. The observation is that the leading digit is 1 about 30% of the time in real-life numerical data sets.
  3. Benford's Law is named after the physicist Benford and is applicable to various sets of numerical data.

Hadamard matrices

30 implied HN points • 03 Dec 23
  1. A Hadamard matrix is an n x n matrix where entries are +1 or -1, with a unique property of rows agreeing in half of their entries.
  2. Hadamard matrices have various applications.
  3. The post on Hadamard matrices is for paid subscribers only.

Automorphic numbers

36 implied HN points • 10 Nov 23
  1. Automorphic numbers are numbers whose square ends with the digits of the number itself.
  2. Examples of automorphic numbers include 625 and 376.
  3. This post is for paid subscribers only.

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.

Toroidal Hitomezashi patterns

48 implied HN points • 12 Sep 23
  1. Hitomezashi is a traditional form of Japanese embroidery with single stitches forming patterns on a grid.
  2. The Toroidal Hitomezashi patterns showcase intricate and beautiful designs.
  3. This post is for paid subscribers only.

The number 163

54 implied HN points • 15 Aug 23
  1. The number 163 is significant in mathematical calculations.
  2. The post discusses the famous constants pi and e.
  3. The number e to the power of pi times the square root of 163 is a close approximation.

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

k-Göbel sequences

54 implied HN points • 21 Jul 23
  1. The Göbel sequence has unique properties, with certain terms being integers and others not.
  2. The 3-Göbel sequence is a variation with even more rapid growth and integer terms.
  3. For any integer k ≥ 2, there is a k-Göbel sequence with its own unique characteristics.

The Thue–Morse sequence

60 implied HN points • 23 Jun 23
  1. The post discusses the Thue-Morse sequence.
  2. The post features an example of a magic square.
  3. Access to the full content requires being a paid subscriber.

Perrin numbers and pseudoprimes

42 implied HN points • 04 Aug 23
  1. The sequence in the picture is called the Perrin numbers.
  2. Perrin numbers follow a specific rule to generate the sequence.
  3. This post is intended for paid subscribers.

The Illumination Conjecture

48 implied HN points • 26 Jun 23
  1. The article discusses the Illumination Conjecture by Richard Green.
  2. The question posed is about how many suns are needed around a planet for an observer to see at least one entirely above the horizon.
  3. The content is available for paid subscribers only.

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.

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.

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.

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.

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

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.

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.

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.