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-Andreev-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.
Although mathematicians have been dealing with polyhedra since ancient times, not everything is known about them. For one thing, every question that is answered only brings about new questions. For example, the number of three-dimensional polyhedra is known only up to 18 vertices. If somebody should find out how many polyhedra there are with 19 vertices, one can immediately ask about the number of polyhedra with 20 vertices. There are also questions that have been waiting a long time for an answer. A „nice“ example, because it is easy to understand and yet still an unsolved problem, is the so-called Dürer conjecture. The painter Albrecht Dürer spent some years studying mathematics and the concept of the net of a polyhedron goes back to him. In his book, „The Painter’s Manual“ he drew nets of several polyhedra.
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.
Siblings:Sibling 1000015, Sibling 1000020, Sibling 1000022, Sibling 1000024, Sibling 1000041, Sibling 1000054, Sibling 1000091, Sibling 1000120, Sibling 1000127, Sibling 1000130, Sibling 1000137, Sibling 1000139, Sibling 1000141, Sibling 1000144, Sibling 1000148, Sibling 1000150, Sibling 1000153, Sibling 1000155, Sibling 1000158, Sibling 1000160, . . .
Sibling 1000162, Sibling 1000166, Sibling 1000170, Sibling 1000183, Sibling 1000185, Sibling 1000186, Sibling 1000187, Sibling 1000195, Sibling 1000196, Sibling 1000197, Sibling 1000202, Sibling 1000206, Sibling 1000227, Sibling 1000232, Sibling 1000235, Sibling 1000249, Sibling 1000302, Sibling 1000305, Sibling 1000306, Sibling 1000310, Sibling 1000358, Sibling 1000428, Sibling 1000452, Sibling 1000454, Sibling 1000455, Sibling 1000459, Sibling 1000482, Sibling 1000488, Sibling 1000491, Sibling 1000497, Sibling 1000522, Sibling 1000525, Sibling 1000528, Sibling 1000530, Sibling 1000535, Sibling 1000542, Sibling 1000544, Sibling 1000545, Sibling 1000550, Sibling 1000552, Sibling 1000554, Sibling 1000555, Sibling 1000563, Sibling 1000577, Sibling 1000606, Sibling 1000616, Sibling 1000696, Sibling 1000717, Sibling 1000719, Sibling 1000733, Sibling 1000737, Sibling 1000769, Sibling 1000781, Sibling 1000782, Sibling 1000825, Sibling 1000832, Sibling 1000841, Sibling 1000844, Sibling 1000848, Sibling 1000850, Sibling 1000853, Sibling 1000854, Sibling 1000855, Sibling 1000859, Sibling 1000861, Sibling 1000866, Sibling 1000878, Sibling 1000885, Sibling 1000893, Sibling 1000898, Sibling 1000901, Sibling 1000922, Sibling 1000925, Sibling 1000954, Sibling 1001011, Sibling 1001014, Sibling 1001017, Sibling 1001037, Sibling 1001052, Sibling 1001071, Sibling 1001117, Sibling 1001118, Sibling 1001129, Sibling 1001149, Sibling 1001160, Sibling 1001162, Sibling 1001164, Sibling 1001166, Sibling 1001170, Sibling 1001173, Sibling 1001175, Sibling 1001177, Sibling 1001210, Sibling 1001212, Sibling 1001213, Sibling 1001222, Sibling 1001232, Sibling 1001344, Sibling 1001350, Sibling 1001352, Sibling 1001355, Sibling 1001368, Sibling 1001371, Sibling 1001376, Sibling 1001380, Sibling 1001383, Sibling 1001391, Sibling 1001401, Sibling 1001404, Sibling 1001440, Sibling 1001463, Sibling 1001470, Sibling 1001475, Sibling 1001479, Sibling 1001490, Sibling 1001497, Sibling 1001500, Sibling 1001528, Sibling 1001566, Sibling 1001568, Sibling 1001570, Sibling 1001580, Sibling 1001591, Sibling 1001594, Sibling 1001725, Sibling 1001732, Sibling 1001734, Sibling 1001745, Sibling 1001759, Sibling 1001761, Sibling 1001763, Sibling 1001766, Sibling 1001768, Sibling 1001770, Sibling 1001771, Sibling 1001774, Sibling 1001775, Sibling 1001777, Sibling 1001779, Sibling 1001782, Sibling 1001785, Sibling 1001789, Sibling 1001791, Sibling 1001793, Sibling 1001795, Sibling 1001807, Sibling 1001821, Sibling 1001838, Sibling 1001847, Sibling 1001851, Sibling 1001885, Sibling 1001899, Sibling 1001901, Sibling 1001913, Sibling 1001984, Sibling 1001996, Sibling 1001998, Sibling 1002075, Sibling 1002116, Sibling 1002125, Sibling 1002130, Sibling 1002132, Sibling 1002134, Sibling 1002141, Sibling 1002143, Sibling 1002160, Sibling 1002162, Sibling 1002163, Sibling 1002165, Sibling 1002167, Sibling 1002171, Sibling 1002176, Sibling 1002178, Sibling 1002181, Sibling 1002184, Sibling 1002186, Sibling 1002187, Sibling 1002189, Sibling 1002192, Sibling 1002197, Sibling 1002199, Sibling 1002201, Sibling 1002209, Sibling 1002212, Sibling 1002246, Sibling 1002248, Sibling 1002253, Sibling 1002260, Sibling 1002276, Sibling 1002281, Sibling 1002287, Sibling 1002294, Sibling 1002296, Sibling 1002297, Sibling 1002302, Sibling 1002303, Sibling 1002306, Sibling 1002350, Sibling 1002352, Sibling 1002393, Sibling 1002398, Sibling 1002401, Sibling 1002415, Sibling 1002449, Sibling 1002455, Sibling 1002458, Sibling 1002461, Sibling 1002469, Sibling 1002475, Sibling 1002478, Sibling 1002480, Sibling 1002482, Sibling 1002487, Sibling 1002490, Sibling 1002494, Sibling 1002496, Sibling 1002524, Sibling 1002571, Sibling 1002576, Sibling 1002578, Sibling 1002584, Sibling 1002588, Sibling 1002590, Sibling 1002592, Sibling 1002595, Sibling 1002598, Sibling 1002600, Sibling 1002602, Sibling 1002608, Sibling 1002615, Sibling 1002618, Sibling 1002625, Sibling 1002628, Sibling 1002630, Sibling 1002633, Sibling 1002649, Sibling 1002660, Sibling 1002685, Sibling 1002686, Sibling 1002701, Sibling 1002731, Sibling 1002756, Sibling 1002781, Sibling 1002785, Sibling 1002787, Sibling 1002788, Sibling 1002797, Sibling 1002799, Sibling 1002801, Sibling 1002816, Sibling 1002818, Sibling 1002819, Sibling 1002824, Sibling 1002841, Sibling 1002845, Sibling 1002846, Sibling 1002851, Sibling 1002853, Sibling 1002857, Sibling 1002859, Sibling 1002862, Sibling 1002866, Sibling 1002868, Sibling 1002875, Sibling 1002878, Sibling 1002879, Sibling 1002886, Sibling 1002887, Sibling 1002890, Sibling 1002897, Sibling 1002901, Sibling 1002903, Sibling 1002905, Sibling 1002908, Sibling 1002910, Sibling 1002913, Sibling 1002919, Sibling 1002922, Sibling 1002938, Sibling 1002942, Sibling 1002945, Sibling 1002948, Sibling 1002958, Sibling 1002959, Sibling 1002962, Sibling 1002964, Sibling 1002967, Sibling 1002974, Sibling 1002976, Sibling 1002978, Sibling 1002980, Sibling 1002983, Sibling 1002987, Sibling 1002989, Sibling 1002991, Sibling 1002995,
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.