The Hypergraphs Around Us

[Türkçe]

A group of things sometimes contains smaller groups (“subgroups”) inside it. Consider the following examples of groups of four things with different types of subgroups:

• In the group of four limbs of the human body, the two arms and the two legs form two separate subgroups.
• Among the four main characters of The Three Musketeers, Athos, Portos and Aramis form a subgroup called the Three Musketeers, while D’Artagnan is left out.
• When juggling four balls using the shower pattern, there are no subgroups – there is no smaller pattern within the main pattern.

This blog post is a collection of groups of things in daily life that have certain subgroups within themselves. The first and most interesting of these collections is the collection of double doubles. You will realize that we can generalize these collections: 4 + 5 + 2 and 9 + 9 + 9 + 1 are not less valid than 2 + 2 or 3 + 1, but the lower sums tend to be more interesting and it was easier to find examples.

At some point I realized that there is a mathematical structure that can be used to describe the categories that I have been sorting the groups into: hypergraphs. A hypergraph is like a standard graph in graph theory, but its (hyper)edges are allowed to connect more than 2 vertices at a time. The categories below (2 + 2, 3 + 1, etc.) are thus various hypergraph isomorphism classes. The hypergraphs that correspond to the categories that I considered for this blog post have the following properties:

• Each edge connects at least two vertices (so, no edges that connect a single vertex to itself) (translation: for a property to create a subgroup, at least two things must share this property).
• There is an edge that connects every vertex in the graph (translation: every thing is related to the rest of the group in some way – there is always some property that all things in the group have in common).

There are many interesting problems related to hypergraphs, but we will not delve into them in this blog post. For those who are interested, hypergraphs show up in machine learning.

2 + 2 (The Double Doubles)

Groups of four things that consist of two pairs. Hypergraph representation:

┏━━━━━━━━┓
┃┏━━┓┏━━┓┃
┃┃AB┃┃CD┃┃
┃┗━━┛┗━━┛┃
┗━━━━━━━━┛

• The first example is also the namesake of this category. Epsilon Lyrae ($\varepsilon$ Lyr) is a multiple star system in the Lyra constellation that consists of two components ($\varepsilon^1$ and $\varepsilon^2$) that orbit each other. Each component is a binary system. Epsilon Lyrae is nicknamed the Double Double.
• Roger Federer has two pairs of twins: two identical boys and two identical girls.
• There are two front wheels and two back wheels in a car.
• I have a friend in college who was doing two joint majors: “Math and Philosophy” and “Computer Science and Economics”.
• Two nunchakus can be used at the same time, one nunchaku in each hand. Each nunchaku consists of two sticks.
• The four cardinal directions can be separated into two pairs: North and South, and East and West.
• The A, S, W, and D keys on a standard English QWERTY keyboard are frequently used as the left, down, up and right buttons in computer games, respectively. Similar to the cardinal directions, they can be grouped as A and D, and W and S.
• In poker, two pair “is a hand that contains two cards of one rank, two cards of another rank and one card of a third rank.” – Wikipedia
• Humans have two arms and two legs. Suggested by Henry.
• Harvard and Yale, Oxford and Cambridge.
• In a doubles match in tennis or in beach volleyball, two pairs of two compete against each other.
• Two couples can go on a double date.
• The four distinct grandparents a person. The mother’s side forms a pair, while the father’s side forms another pair.
• In a single-elimination tournament, there are two semifinal matches, each of which pair two players/teams.
• SATB is an initialism that describes the scoring of compositions for choirs or consorts of instruments. The initials are for the voice types: S, soprano, A, alto, T, tenor and B, bass. Soprano and alto are female voice types (pair #1), while tenor and base are male voice types (pair #2).
• Most US presidential races are essentially between two teams: 1) Democratic presidential and vice presidential candidates, and 2) Republican presidential and vice presidential candidates.
• There are four main sections of a human heart: the right atrium and the right ventricle, and the left atrium and the left ventricle.
• Lobsters have two long antennae and two smaller antennules. Suggested by Raymond.
• Some animals like the moray eel have two sets of jaws: the oral/primary jaws, and the pharyngeal jaws. Suggested by Raymond.
• A dragonfly has two pairs of wings. Suggested by Raymond.
• “Double-Double” is a burger in In and Out Burger that has two patties and two slices of cheese (there is also a “triple double” menu item?).
• In some pistol duels, each team consists of a dueler and their second. Suggested by Zach.
• An alpha particle (or equivalently, the nucleus of a Helium-4 atom) consists of two protons and two neutrons.
• The electron configuration of a neutral Beryllium atom: $1s^2 \ 2s^2$.
• “Tropomyosin is a two-stranded alpha-helical, coiled coil protein found in many animal and fungal cells” – Wikipedia. Suggested by Efe.
• In the 4-balled version of fountain, a juggling technique, each hand juggles two balls, and every ball stays in the same hand.
• Each foot gets a sock and a shoe.

The ones below are somewhat close but don’t fit the list:

• In music, four quarter notes that come in pairs (two notes pressed at a time), and the two pairs are connected from above with a beam.

2 + 1

Groups of three things that consist of a pair and a third one. Hypergraph representation:

┏━━━━━━━┓
┃┏━━┓┏━┓┃
┃┃AB┃┃C┃┃
┃┗━━┛┗━┛┃
┗━━━━━━━┛

• The 1992 US presidential election was mainly fought between Clinton (Democrats) and Bush (Republican), and the third-party candidate Perot.
• In a deuterium atom, a proton and a neutron form the nucleus, while an electron orbits around them.
• A water molecule ($H_2O$) consists of two hydrogen atoms and an oxygen atom.
• In Indonesian, you say the word two times to make it plural. The plural of a two-word noun phrase in Indonesian is made by writing the first word twice and the second word once.
• Some animals have a nictitating membrane, which is also called a third eyelid.
• In a semester system, the year is split into two semesters and the summer.

3 + 1

Groups of four things that consist of a group of three things, and a fourth one that doesn’t fit the group of three. Hypergraph representation:

┏━━━━━━━━┓
┃┏━━━┓┏━┓┃
┃┃ABC┃┃D┃┃
┃┗━━━┛┗━┛┃
┗━━━━━━━━┛

• In tennis, the Big Four consists of Rafael Nadal, Roger Federer, Novak Djokovic and Andy Murray. The Big Three excludes Murray.
• D’Artagnan and the Three Musketeers (Athos, Porthos and Aramis) makes a group of four.
• Some of the newest smartphones (like the latest of the Samsung Galaxy series) have three cameras on the back, one camera on the front.
• The Hobbit is the prelude to The Lord of the Rings, which is a trilogy. Suggested by Zach.
• Galatasaray, Fenerbahçe, Beşiktaş and Trabzonspor are the four most popular football teams in Turkey, while the first three are considered to be the big three.
• In a quarter system, the year is split into three non-summer quarters, and the summer.
• The Fire Diamond consists of three numbers and a symbol.