Geometry explains a large part of the world around us. It not only describes what we can sense but also what we may imagine. As its own branch of mathematics, it is concerned with questions of shape, size, the relative position of figures, and the properties of space.

Historically, basic geometric shapes (e.g. square, circle) are found in a number of early cultures (e.g. Mayan and Mesoamerican cultures). Such shapes were first considered since they can be created by combining specific amounts of lines and/or curves. The consideration of more complex 3D shapes seem to have permitted mathematics and more broadly science to be what it is today. An important example dates back from ca. 2000 BC in prehistoric Scotland, where carved stone balls exhibit a variety of symmetries, including all of the symmetries of Platonic solids. Another noteworthy piece of evidence is from 1800 BC, in Ancient Egypt: the Moscow mathematical Papyrus. It shows that calculating areas and volumes of various geometric shapes such as a cylinder and pyramid was already feasible.

Geometry inspired many designers across different cultures. In the particular example of the circle, this spans from compositions obtained from circles, disks, and parts of them in the Key of Solomon (ca. 1350), to I’itoi, or the man in the maze from Native American culture, to Marcel Duchamp’s optical disks or rotoreliefs (1936). In the early 1960s, a visual case study on these basic shapes was published by Bruno Munari.

Definitions

Mathematically, these basic shapes are referred to as polygons. A polygon is a plane figure with at least three straight sides and angles. A regular polygon is a many-sided polygon in which the sides are all the same length and are symmetrically placed about a common center, i.e. the polygon is both equiangular and equilateral.

Using these polygons, we can define a polyhedron as a solid figure with many plane surfaces (faces) and intersections (vertices).

• A convex polyhedron can be defined as a polyhedron for which a line connecting any two (non-coplanar) points on the surface always lies in the interior of the polyhedron.
• A regular polyhedron can be defined as a polyhedron for which every face is the same regular polygon, and for which every vertex is regular.

The 92 convex polyhedra having only regular polygons as faces are called the Johnson solids, which include the Platonic solids and Archimedean solids.

Platonic solids

In the particular case of 3D shapes where each face is the same regular polygon and the same number of polygons meet at each vertex or corner, the polyhedra are highly symmetrical. This means their appearance is unchanged by reflection, or rotation of space. These special polyhedra are both regular and convex. They are named after Plato: Platonic solids. As indicated by the longest mathematical proof to date on “the classification of finite simple groups,” there are only five Platonic solids: tetrahedron, hexahedron, octahedron, dodecahedron, icosahedron. For example, a hexahedron, also known as a cube, consists of six same-sized square faces, and every three squares meet at each vertex.

Each polyhedron is displayed as a polyhedral graph, or the undirected graph formed from the vertices and edges of a convex polyhedron.

Archimedean solids

The faces of Archimedes polyhedra are regular polygons of two or three types. Their vertices are superimposable, irregular, and lay on a circumscribed sphere. Their edges have the same length.

Now imagine if we were to cluster a series of spheres of the same size like a cluster of graphs. To do so, we subject them to equal and steady pressure by submerging them in water, little pressure areas would build up between the spheres. If such pressure is permitted to build up, the spheres will eventually collapse into their more stable shape: a cluster of tetradecahedra.

Tetradecahedron

A tetradecahedron is one of the 13 polyhedra of Archimedes. It was discussed in the 17th century by Kepler. It is sometimes called the cuboctohedron (Thomson, 1887). It has exactly 13 vertices: a central one, plus the outer ones that define 12 directions in space. As shown below, there are 12 elements in the neighborhood of the central element arranged in four interleaved hexagons. It is also a semi-regular polyhedron since its 14 sides or faces are different polygons.

It is the only solid in which the length of the vertices is equal to that of the radial distance from its center of gravity to any vertex. This implies that the line drawn from the center of this polyhedron is the same length no matter the direction, making it rounder than a cube but squarer than a sphere.

When the tetradecahedron’s vertices are represented as equidistant spheres and this representation is projected onto two-dimensions, we obtain the Metatron cube. The latter figures in Kabbalist scriptures and Christian art.

Patterns

The repetition of certain shapes creates intricate patterns. Some of these repetitive patterns are historic ornamental art and have been reported across different cultures.

Some of these patterns look strikingly similar to artistic designs of various styles. This reflects not so much a similarity in underlying rules, but rather similarity in features that are most noticeable to the human visual system.

Today, we could use a set of rules to computationally create such patterns, for example, by using a cellular automaton. Note that square grids of colored cells as in the cellular automata can be used quite directly as weaving patterns in textiles.

Cellular automata

A cellular automaton is a collection of “colored” cells on a grid of specified shape that evolves through a number of discrete time steps according to a set of rules based on the states of neighboring cells. The rules are then applied iteratively for a given number of steps.

The best-known cellular automaton is John Horton Conway’s Game of Life, also known as the Game of Life or simply Life, created in 1970. As seen in the example Game of Life, the “game” is actually a zero-player game.

This means its evolution is determined by its initial state, needing no input from human players. The only interaction is creating an initial configuration and observing how it evolves. You may try it out here.

Cellular automata are used to simulate various phenomena, from gas flows to pedestrian traffic in stations and other physical, chemical, or sociological problems.

A noteworthy example is reproducing the wave patterns on the skin of cephalopods (such as a squid, or octopus) which corresponds to the activity of chromatophores. This can be simulated with a two-state, 2D cellular automata, in which each state corresponds to either an expanded or retracted chromatophore.

Some patterns remain unchanged after one iteration or period. These patterns are referred to as life forms or still lifes. They have been extensively documented by Eric Weisstein. By definition, a pattern which does not change from one generation to the next is known as a still life. Several still lifes are illustrated. However, if a life pattern does not have a father pattern, it is called a Garden of Eden for obvious biblical reasons (creationism).

A Garden of Eden is a term that was first used in connection with cellular automata by John W. Tukey, many years before the Conway’s Game of Life — the best-known cellular automaton — was conceived.

Systems like cellular automata can readily be set up on any geometrical structure in which a limited number of types of cells can be identified, with every cell of a given type having a similar neighborhood. In a simple case, the cells of a system are all identical and are laid out in the same orientation in a repetitive array. This means that all cell centers form a lattice.

When considering the hexagonal kernel of Smith (1969) and extending it into three dimensions, the result is the tetradecahedron. This extension is a logical transform. In this case, it also corresponds to a hexahedral tessellation (i.e. an arrangement of polygons in a repeated pattern without gaps or overlapping). Such an extension of a two-dimensional logical transformation to spaces of higher dimensionality provides a practical usage in image processing.

Image processing and analysis

Although image processing and analysis is often seen as operations on 2D arrays of values or images, there is however a number of fields where images of higher dimensionality must be analyzed. The analysis of image data from biological tissue, in both Biology and Medicine, requires such operations. Common operations are correlations, convolutions, and filter functions. Computationally, multidimensional filters are implemented as a sequence of one-dimensional filters.

Cell packing

In structural design and materials science, appropriate models are required to design microstructure components that make up a strong and stable structure. As the cell packing terminology suggests is, cells are compact and the way they are packed determines how robust the overall structure is. A cell may be shaped like a number of polyhedra.

As an example, closed-cell foams are made up of cells that are, completely closed, and are pressed together, so air and moisture are unable to get inside the foam. The most important foam model is Kelvin’s tetradecahedron, which has been derived from the idea to construct a spatially repeating regular cell-packing model with a minimum surface-to-volume ratio.

The aforementioned tetradecahedron is used in the design of porous metal foams. In the context of effective thermal conductivity, a highly porous aluminum foam has been developed (Yang 2014). This porous cellular aluminum structure consists of inner-connected pores and fiber-made skeleton. Due to the connectedness of its inner pores, the foam can be saturated by a fluid (e.g. air, water).

Although the space-filling shape of a tetradecahedron is used in the design of polyester sponge foams to maintain minimal surface energy (Thomson 1887, Weaire and Hutzler 2001) and a minimum total face area (Kelvin 1887), other research suggests that other designs can be more efficient. In 1993, Denis Weaire and Robert Phelan discovered a layered repetitive arrangement of 12- and 14-faced polyhedra that yields 0.003 times less total area. Moreover, it is possible that there are other polyhedra which fill space in a less regular fashion and yield still a smaller total area. In the special case of maintaining the minimal surface area, the polyhedra are slightly curved.

Cell packing also occurs naturally. Circa 1923, Lewis showed that cells of the elder pith (subfigure B), cells of the human fatty tissue, and skin cells or epidermal cells have on an average fourteen faces, and therefore are or tend to be tetradecahedra. This was also described in 1979 for the inner-wall cells of a sunflower.

In the case of mammalian epidermal cells, researchers observed — using in vivo 3D imaging — that they do follow the cell-packing model, in which the cells have a flattened tetradecahedron shape.

It also happens that some crystals use a tetradecahedron as a space division technique: Magnetite (Fe3O4), Magnesium silicide (Mg2Si), and other crystalline selenides: Ta2P, NiTa8Se8. In the context of crystal growth, these tetradecahedra based models are reproducible and help in modeling new materials. Yet they do not account for the geometric uncertainty of the microstructure. To circumvent this problem, a Voronoï process generates cell structures representative of the microstructure.

If you’ve read this far, you might have noticed a recurrent mention of the tetradecahedron. Thanks to its particular properties, it’s used in various fields and serves as a great example of the extent of Geometry. Moreover, it’s found now technologically in Space (with a big s) and aeronautics. Imagine it’s the shape of a space module — that is a standardized part or independent unit — we can construct a more complex structure. Provided the functionality of such modules is solved for living quarters, clustering them, and shaping cities on other planets is a logical step.

In the previous sunflower example, it is also commonly known that the flowers show a Fibonacci sequence. In 2016, a study by Swinton et al showed that it’s not as simple as everyone thought. Results indicated that nearly one in five of the flowers had either non-Fibonacci spiraling patterns or patterns more complicated than has previously been reported.

Vincent Van Gogh couldn’t stop reiterating the sunflower and his attachment to the subject and its study can be felt. Although there are other paintings with “Nothing but large Sunflowers”, Van Gogh was completely enthralled by sunflowers.

Since humans are a pattern-seeking species, geometry helps us to shed light over curious occurrences. Some forms are considered hypnotizing and sacred (e.g. the flower of life—top left figure below). Ultimately, geometry answers some interesting questions.

Here’s another piece in the series: