Polyhedra are geometrical bodies that consist of vertices, straight edges and flat faces. In our project we restrict ourselves to convex polyhedra. This means that all inner angles between two edges or two faces are less or equal to 180°. No cavities, holes or indentations are allowed.
The Platonic solids are a very symmetric and regular class of five polyhedra. They consist only of congruent (same shape and size), regular (all sides the same length) polygons. The tetrahedron, the octahedron and the icosahedron comprise of regular triangles. In every corner of the tetrahedron, three triangles meet. In the case of the octahedron four and with the icosahedron five triangles build a joint at each vertex. If you put six equilateral triangles at a vertex, which each have an interior angle of 60°, you get a full circle of 360° and they lie flat on the ground. Therefore, no new body emerges. Similarly, the cube is built from three squares and the dodecahedron from three equilateral pentagons at each vertex. Euclid used this approach in his proof to show that there can be no further polyhedra with these properties, thus classifying the five Platonic solids.
But symmetric polyhedra are not the only interest of mathematicians. For their research they are searching for polyhedra (or their higher-dimensional equivalent, the polytopes) that have specific properties. We have asked some geometers – this is what the mathematicians who study geometry are called – about their favorite polyhedra and this is what they said:
“If polytopes could be viewed as rocks, then the associahedron is the diamond of polytopes. Diamonds are made of a very common element in nature–carbon–and likewise the associahedron can be realized via very common tools. Yet it enjoys such a unique and rare structure–and provides such a fascination–that no other polytope may ever be compared to the diamond.” - Jean-Philippe Labbé.
“My favorite 3-polytope is the icosahedron, for its complexity yet simplicity. If you start gluing equilateral triangles, five at a vertex, and no matter what you do you end up with this nice thing. When I got bored in high school (which, yes, happened), I drew icosahedra in the margins in my notebooks. Sometimes they were not totally regular; I amused myself making them look like faces. Last but not least, I am fascinated by the fact that you can decompose its 12 vertices into three golden rectangles intertwined as Borromean rings.” - Francisco Santos.
“My favourite polyhedron is “Miller’s solid”, also known as the ”pseudo-rhombicuboctahedron“ or as the ”elongated square gyrobicupola“, probably first found by D. M. Y. Sommerville in 1905 — an object that was often overlooked (already by Archimedes?), discovered and rediscovered (by J. C. P. Miller, among others). It is pretty, but if you look closely it has a certain twist, so it is not perfect. It looks classical, like an Archimedean solid, but it isn’t really, at least according to the modern definition of an Archimedean solid. Thus it is a good reminder that we have to be careful with definitions in mathematics, and always look at possible exceptions and special cases.” - Günter M. Ziegler.
„My favourite polyhedra are the Koebe polyhedra. All their edges touch a sphere. All faces of these polyhedra have inscribed discs. The discs of neighbouring faces touch. These exist an explicit dualization procedure that generates discrete minimal surfaces from Koebe polyhedra. The corresponding surface is then a discrete P-Schwarz surface and the Koebe polyhedron is its Gauss map. One can read more in: A.I. Bobenko, T. Hoffmann, B.A. Springborn, Minimal surfaces from circle patterns: Geometry from combinatorics, Ann. of Math. 164:1 (2006) 231-264) and can watch the movie: (http://discretization.de/en/movies/koebe/).“ - Alexander Bobenko.
A polygon is a two-dimensional polyhedron which consists of vertices and edges. The area bounded by the edges is the polygon itself. Three-dimensional polyhedra are made up of vertices, edges and polygons, i.e., the faces.
A special class of polygons are called the regular polygons. They are made up of equilateral edges (all the same length) and all their inner angles are the same. Some examples of regular polygons are the square, equilateral triangle, and equilateral pentagon. Regular polygons form the building blocks for the Platonic and Archimedean solids.
If you cut a hollow cube on enough edges, unfold it and lay it flat on the plane, you get what is called the net of the cube. If you draw this net on paper, you get an outline which you can cut out and glue together to form the cube. Of course, this method works for any polyhedra. In our project, we use exactly these templates to build our models. We simulated this process of cutting open and unfolding polyhedra with a computer and automatically created the nets of the polyhedra.
A cube with its net and a polygon with seven corners and the associated net.
Another very symmetric and hence “beautiful” class of polyhedra are the Archimedean Solids. They also only consist of regular polygons, but here unlike the Platonic Solids a combination of them is allowed. The most common representative is the soccer ball. Mathematicians would rather speak of a truncated icosahedron, since it emerges when you chop off the tip of each vertex of an icosahedron.
On the left an icosahedron and on the right its truncation - a soccer ball.
How many polyhedra are there?
For every fixed number of vertices, there are a certain number of polyhedra. In the table, the number of different types of polyhedra is given for the number of vertices. It is clear that the number of types increases rapidly. If you have four points in space, they are either all on the same level (not three-dimensional), or the shape will be a pyramid over a triangle. Therefore, there is only one polyhedron with four vertices, the tetrahedron.
For five vertices, there are two possibilities: the pyramid over the square, if four of the five vertices lie the same plane, or the double pyramid over a triangle. For six vertices, finding the seven different types starts to get more complicated.
In order to find out how many polyhedra types actually exist for each number of vertices, we have to create and list them. But how do you know that this list is complete and no polyhedron is counted twice? In geometry, Steinitz’s Theorem states that each polyhedron can be uniquely assigned to a graph with certain properties. (Here the notion of a graph is not referring to the ones living in coordinate systems but the ones that are subject to graph theory). These graphs are mathematically easier to grasp and therefore count. But even for this, you will need a computer because the numbers get very large very fast. The number of seven- and eight-vertice polyhedra, 34 and 257 respectively, were found back in 1899. For the discovery of the 2606 nine-vertice polyhedra in the year 1969 the invention of the computer was necessary.
In mathematics, there are many ways to interpret dimensions. One way is to imagine dimensions as the number of variables. For example, the ingredients of an apple pie (flour, butter, sugar, eggs, baking soda and apples) can be understood as six variables and therefore the apple pie is a six-dimensional object.
By looking at photos and films, which are a representation of our three-dimensional world in a two-dimensional medium, we are used to seeing an extra dimension. This process of mapping a higher dimension into a lower one, that is taking a two-dimensional photograph of the three-dimensional world, is called a projection in mathematics.
Unfortunately, it is not possible to truly represent four-dimensional space in the three-dimensional space surrounding us, but we can use projections to understand it. For example, if you look at a cube, its faces are squares. A square can be thought of as a two-dimensional cube, because all its sides are the same length, so the sides of the three-dimensional cube are two-dimensional cubes. This idea extends to higher dimensions. The side surfaces of a four-dimensional cube are three-dimensional cubes. The result is a so-called tesseract. Here is a link to a video where this relationship is graphically visualized.
When we speak of polyhedra, we silently assume that they are convex polyhedra. Convex means that there are no indentations, cavities or holes. The mathematical definition of convexity states that for any two points that lie within a set, a straight line connecting them must lie completely within the set.
On the left a convex, on the right a non-convex object.
Each polyhedron can be geometrically realized in different ways. It can be big or small, and its shape can also be changed, as long as the structure of the vertices, edges and surfaces remains the same. This structure, which is the number of edges meeting at the vertices, and the number of vertices belonging to each surface, is called the combinatorial type of a polyhedron. We call two polyhedra combinatorially equivalent if they possess the same combinatorial type, i.e., one can uniquely assign vertices to each other so that if two vertices in one polyhedron are connected by an edge, then the vertices in the other polyhedron are connected by an edge. Every polyhedron has an infinite number of different geometric interpretations. If you choose a polyhedron on Polytopia.eu, you will adopt the entire combinatorial type. So you have actually adopted infinitely many polyhedra. To make it less confusing and easier to make the model, we have chosen a clear realization of the polyhedron. These are the so-called Koebe-Adreev-Thurston realizations of polyhedra. In particular, these realizations have a sphere inscribed inside the polyhedra that touches each of the edges at exactly one point. In particular, each surface contains a circle that touches the edges just once.
The f-vector of the polyhedron indicates how many vertices, edges, and faces it has. A vector in this case is not a geometric quantity but only the way of representing these numbers. The cube consists of 8 vertices, 12 edges, and 6 faces, and thus has the f-vector (8,12,6). However, the polyhedra are not uniquely determined by this vector. There may be other polyhedra with the same f-vector that have a completely different structure. We call these polyhedra siblings.
Here we see a cube and its sister. She also possesses 6 faces, 12 edges, and 8 vertices but contains an entirely different structure that the cube.
Physical models and their construction have long played an important role in mathematics. For one thing, there was simply no other way to understand ideas in a three-dimensional environment. Of course, three-dimensional models can always be drawn, but then the drawing is only a projection of the model onto the plane, much like taking a picture of the model. When it comes to photos of familiar objects, recognizing the space does not cause us any problems, because we know, for example, that a table is usually right-angled. If we see a perspectively distorted table in a photo, we intuitively know about the right angles. Of course, this intuition is not there when trying to understand the structure of an unfamiliar geometric object. In order to recognize certain properties, such as an axis of symmetry, it is very helpful to actually hold an object in your hand and turn it.
Models serve not only to gain knowledge but also to share knowledge. To make their research accessible to others, mathematicians needed a way to visualize it. Nowadays, this is done mainly with computers. There is a lot of software to generate mathematical and geometric graphics. Rotation of a model using this software also counteracts the problem of restriction to the flat screen.
Obwohl sich schon seit der Antike Mathematikerinnen und Mathematiker mit Polyedern beschäftigen, ist noch lange nicht alles über sie bekannt. Ein „schönes“, weil leicht verständliches und doch bislang ungelöstes Problem ist die sogenannte Dürer’sche Vermutung. Der Maler Albrecht Dürer hat sich einige Jahre lang intensiv mit Mathematik beschäftigt. Auf ihn geht auch das Konzept des Polyedernetzes zurück. In seinem Buch „Underweysung der Messung, mit dem Zirckel und Richtscheyt“ zeichnete er Netze von verschiedenen Polyedern.
A net of a polyhedron is created by considering the polyhedron as an empty shell, which is cut along its edges in such a way that it remains connected but can be laid flat without distorting the faces. The question behind Dürer’s conjecture is whether this is possible for every polyhedron such that the faces do not overlap when it is unfolded. In other words, does every polyhedron have a net?
To date, many mathematicians have considered this question and there are some intermediate results. For example, it is known that you can unfold any polyhedron without overlap if you pull and distort the faces and thus change its geometrical realization but not its structure (s. https://arxiv.org/pdf/1305.3231.pdf). The only polyhedron for which we know for sure that can always be unfolded without changing its geometrical structure is the tetrahedron.
Since we have automatically generated the unfoldings for the polyhedra in our project, it is possible that the net of your adopted polyhedron is overlapping. If so, write us an email!
Polyhedra are siblings if they contain the same number of vertices, edges and faces, hence the same f-vector. Similar to human siblings, some polyhedral siblings do look alike each other while others have a completely different form. The cube consists of 8 vertices, 12 edges and 6 faces. These numbers do not uniquely define its structure. There are polyhedra who have the same f-vector but an entirely different structure.
Fields of Application of Polyhedra
From a purely mathematical perspective, polyhedra are, above all, beautiful and interesting, and their exploration requires no further justification. Nevertheless, one can obviously ask the question, which is almost as old as mathematics itself, what do you really need it for?
One important application of polyhedra is Linear Optimization. It is a method that is often used in business, among other areas, to make decisions that depend on many factors.
One example is making a timetable and network line for a public transportation system. There are many variables to be considered, such as arrival and departure times, operational costs, line capacities and so on. City planners want to meet public expectations for how often a train comes and also minimize the costs, run enough trains to carry enough passengers, but are also limited to the number of trains on the tracks for safety reasons. From these variables, a system of linear inequalities arise and their set of possible solutions form a polytope. The optimal solutions are located at the vertices of that polytope. So finding these vertices gives city planners optimal ways to build the most effective timetable possible.
How do the names of mathematical objects actually come about?
The Greek word for five is “penta”, so a pentagon is a five sided polygon. The hexagon, heptagon, and octagon get their names in the same way, but there is no trigon. Instead, a triangle is the 3-sided polygon and gets its name from its three angles. But then what is a square? Clearly, it is not enough that the name alone can give a definition. Although “square” does not describe the features, it is a commonly known shape. Therefore, it is necessary to actually use the name so that its meaning is well known.
Mathematical objects are also often named after mathematicians. More often than not, these objects and other concepts have been named after male mathematicians, but female mathematicians have also left a legacy behind. The Noetherian rings, named after Emmy Noether, and the Witch of Agnesi, after Maria Agnesi are some examples, but there is a need to close the gender gap.
Mostly, the objects that are named after mathematicians are given these names by scientists. The concept of a ring was already known, but to be able to distinguish the rings that Emmy Noether wrote about from the general ones, one talked about Noetherian rings. The convention of these rings came first and later a definition was established.
The Dürer conjecture was never proposed by the painter Albrecht Dürer himself but the underlying nets of polyhedra go back to him. The conjecture itself was posed by the mathematician G. C. Shephard in 1975. Why then it is known as Dürer’s and not Shephard’s conjecture, one can only speculate.
In summary, the rules and conventions for naming are rather ambiguous. It is similar to getting a nickname – if everyone knows who or what is meant, then the name sticks.