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

Rubik’s abstract polytopes

18 implied HN points • 03 Mar 25
🔬 Science Mathematics Topology Geometry Puzzles Education
  1. Rubik's Cube can be made in different shapes, like a tetrahedron or dodecahedron, instead of just the classic cube. These variations have their own names, like the Megaminx for the dodecahedron.
  2. A new study explains how to think about these puzzles in more dimensions, counting how many different ways they can be turned or rearranged. This includes understanding shapes like the hypercube.
  3. The math behind a Rubik's Cube shows interesting patterns, like counting sticks and pieces, and this can help us understand all kinds of shapes and designs better.

Counting dimer tilings

36 implied HN points • 21 Feb 25
🔬 Science Mathematics Combinatorics Physics Statistics Theory
  1. Dimer tilings involve arranging domino-shaped pieces on grids, and how many ways you can arrange them can vary based on the layout. For example, on a 3x3 grid with one space empty, there are 18 different arrangements.
  2. If at least one dimension of a rectangle is even, it's possible to cover it completely with dimers. However, if both dimensions are odd, it's impossible to cover them without leaving gaps.
  3. There are mathematical patterns and theorems, like Gomory's Theorem, that help understand how to tile grids with dimers. These principles can show when tiling is possible based on the arrangement and color of squares.

The maximum number of SETs for 12 cards

90 implied HN points • 10 Feb 25
🔬 Science Mathematics Linear Algebra Game Theory
  1. The game SET uses 81 cards that have four qualities: quantity, shading, color, and shape. Players look for sets of three cards where each quality is either all the same or all different.
  2. SET can be understood through linear algebra, where each card is represented as a four-dimensional vector. If the vectors for three cards add up to zero, they form a valid set.
  3. Recent research showed that with 12 cards, a maximum of 14 sets can be formed, and they provided proofs for similar results with fewer cards. This reveals interesting mathematical properties of the game.

Identifying bottlenecks in networks

48 implied HN points • 03 Feb 25
🔬 Science Mathematics Networks Physics Statistics Logic Analysis
  1. Bottlenecks in networks are crucial points that can slow down communication or movement. Identifying these points helps understand how the entire network functions.
  2. Networks can be made up of different regions that are linked by these bottlenecks. Recognizing connections between these regions is important for overall analysis.
  3. Knowing where the bottlenecks are can help improve the efficiency of networks, whether in transportation or social connections. This can lead to better planning and resource allocation.

Turning a triangle into a square

115 implied HN points • 11 Jan 25
🔬 Science Mathematics Geometry Puzzles Research
  1. Henry Dudeney showed in 1902 that you can cut an equilateral triangle into four pieces and rearrange them into a square with the same area. This is a fun example of how shapes can transform while keeping their total area the same.
  2. The Wallace–Bolyai–Gerwien theorem explains how you can rearrange two shapes with the same area into each other through cutting, but Dudeney's method is unique because the pieces stay connected during the transformation.
  3. Recent research proved that you can't turn a triangle into a square using fewer than four pieces without flipping any. This shows how specific and tricky these geometric dissections can be.
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Chess and the number e

163 implied HN points • 16 Dec 24
🔬 Science Mathematics Combinatorics Statistics Game Theory Logic
  1. The number e, around 2.718, plays a big role in math, especially in combinatorial problems like derangements. This is when items are arranged so that none are in their original position.
  2. In chess, setting up nonattacking rooks can be related to derangements. The chance that none of them land on the main diagonal equals about 36.8%, which links back to the number e.
  3. Recent studies have also looked at how many safe squares remain on a chessboard when placing random pieces. As more pieces are added, the proportion of safe squares follows certain patterns connected to e.

The mathematics of the game Waffle

48 implied HN points • 22 Jan 25
🔬 Science Mathematics Game Theory Algorithms
  1. Waffle is a fun word game where you need to form six five-letter words in a grid. You can swap letters to find the right words based on clues given.
  2. To solve Waffle, you must figure out the words first, then how to rearrange the letters, and finally do it using the least number of swaps.
  3. Group theory is useful for solving Waffle puzzles because it helps to find ways to rearrange the letters efficiently, especially when dealing with repeated letters.

Constructing space-filling curves

90 implied HN points • 30 Dec 24
🔬 Science Mathematics Physics Biology Chemistry Engineering
  1. Space-filling curves, like the Hilbert curve, can fill a whole area by connecting points in a specific way through iterations. They start small and grow by adding more points and connections at each step.
  2. Different seeds can lead to different types of curves. Each seed can be developed using two choices for how to connect the points, leading to many possible variations.
  3. The process used to create these curves can also be reversed. By looking at a curve and breaking it down, you can see how it was made step by step.

The Ramanujan Machine

90 implied HN points • 23 Dec 24
🔬 Science Mathematics Technology Artificial Intelligence Research
  1. Srinivasa Ramanujan was a brilliant mathematician known for his unique insights and identities, many of which he discovered in unconventional ways.
  2. The Ramanujan Machine is an AI project that helps generate new mathematical conjectures, making it easier to discover complex equations related to fundamental constants.
  3. The odd double factorial is a useful concept in pairing problems and can be calculated by multiplying all odd numbers up to a certain point, making it easier to understand how to pair off groups.

Chutes, ladders, and Markov chains

72 implied HN points • 04 Dec 24
🚌 Education Mathematics Statistics Game Theory Probability Teaching
  1. The game of Chutes and Ladders is a fun example of a Markov chain. It shows how the next move depends only on where you are now, not on how you got there.
  2. There are different types of game boards, some allow for winning while others can trap players forever. Ultimately winnable boards guarantee that a player can reach the end if they keep playing.
  3. On average, players need about 39 spins to win the game, and surprisingly, most random boards created will still offer a winning chance.

A polynomial with Rubik’s cube symmetry

78 implied HN points • 25 Nov 24
🔬 Science Mathematics Symmetry Group Theory
  1. Rubik’s Cube has a huge number of ways it can be scrambled, around 43 quintillion, which shows its interesting symmetry in math. It can be thought of as not just a puzzle, but a complex mathematical object.
  2. There are specific rules about how the pieces of the Rubik’s Cube can be rearranged, which creates a lot of interesting patterns and symmetries. This helps mathematicians understand how groups of movements relate to each other.
  3. Recent research has shown that it's possible to find polynomials that have the same symmetries as the Rubik’s Cube. This connects the world of puzzles to deeper mathematical concepts, making it a fun area to explore.

Repunits and prime numbers

42 implied HN points • 18 Nov 24
🔬 Science Mathematics Number Theory
  1. Repunits are numbers made only of the digit 1 and can appear in different bases. For example, the number 31 can be written as 111 in base 5 and base 2.
  2. Mersenne primes are special numbers of the form 2^p - 1 that can be prime, where p is also a prime. However, it's rare for these to actually be prime numbers.
  3. One interesting link is between Mersenne primes and perfect numbers, which are those that equal the sum of their divisors. Each Mersenne prime corresponds to a perfect number, like how 31 corresponds to the perfect number 496.

The Parks puzzle

36 implied HN points • 11 Nov 24
🚌 Education Mathematics Puzzles Computer Science Logic
  1. The Parks puzzle is a game where you place trees on a grid with specific rules, similar to Sudoku. Each row, column, and park needs a certain number of trees without them being next to each other.
  2. While checking if a proposed solution is correct is easy, finding that solution can be quite complex. Researchers found that the Parks puzzle belongs to a group of difficult problems called NP-complete.
  3. The puzzle can be used to model logical operations like AND and OR. This means it has connections to computer science concepts and can help explore complex problems.

Hexagonal knot mosaics

24 implied HN points • 03 Nov 24
🔬 Science Mathematics Geometry Topology Research
  1. Hexagonal knot mosaics are a way to represent knots on a hexagonal board. You can use different types of tiles to create them.
  2. There are three categories of hexagonal mosaics: standard, semi-enhanced, and enhanced. Each type has different rules about how crossings can occur.
  3. Research has shown the maximum number of crossings you can achieve in these mosaics. Enhanced mosaics can have the most crossings, while standard ones allow the least.

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.

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

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.

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.

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.

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.

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.

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

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.

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.

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.

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.

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.

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.

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.

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 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.

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.

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.

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.

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.

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.

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.

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 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.

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.