## Glossary

### Polyhedron

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 most prominent examples of polyhedra are the cube and the pyramid. You also may have encountered the prism or the octahedron. But there are so many more polyhedra.

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.

### Polygon

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.

### Polyhedral Nets

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.

### Archimedian Solids

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.

Ecken |
Polyeder |

4 | 1 |

5 | 2 |

6 | 7 |

7 | 34 |

8 | 257 |

9 | 2.606 |

10 | 32.300 |

11 | 440.564 |

12 | 6.384.634 |

13 | 96.262.938 |

14 | 1.496.225.352 |

15 | 23.833.988.129 |

16 | 387.591.510.244 |

17 | 6.415.851.530.241 |

18 | 107.854.282.197.058 |

19 | ??? |

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.

### Dimension

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.

### Convex/Convexity

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.

### Combinatorial Type

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.

### f-Vector

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.

### Mathematical Models

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.

### Dürer’s Conjecture

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!

### Siblings

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:Polytorsten, Pollister , Rain, PengPeng, Sibling 1000009, Sibling 1000013, Sibling 1000014, Sibling 1000017, Sibling 1000019, Sibling 1000023, Sibling 1000025, Sibling 1000029, Sibling 1000031, Sibling 1000040, Sibling 1000042, Sibling 1000044, Sibling 1000045, Sibling 1000049, Sibling 1000053, Sibling 1000078, . . .
Sibling 1000085, Sibling 1000090, Sibling 1000093, Sibling 1000116, Sibling 1000119, Sibling 1000123, Sibling 1000126, Sibling 1000129, Sibling 1000132, Sibling 1000134, Sibling 1000136, Sibling 1000140, Sibling 1000142, Sibling 1000143, Sibling 1000152, Sibling 1000157, Sibling 1000159, Sibling 1000161, Sibling 1000165, Sibling 1000169, Sibling 1000171, Sibling 1000174, Sibling 1000176, Sibling 1000179, Sibling 1000180, Sibling 1000182, Sibling 1000184, Sibling 1000189, Sibling 1000190, Sibling 1000194, Sibling 1000199, Sibling 1000200, Sibling 1000201, Sibling 1000204, Sibling 1000205, Sibling 1000216, Sibling 1000218, Sibling 1000220, Sibling 1000225, Sibling 1000226, Sibling 1000229, Sibling 1000231, Sibling 1000234, Sibling 1000248, Sibling 1000253, Sibling 1000262, Sibling 1000263, Sibling 1000265, Sibling 1000268, Sibling 1000281, Sibling 1000300, Sibling 1000301, Sibling 1000304, Sibling 1000308, Sibling 1000309, Sibling 1000312, Sibling 1000316, Sibling 1000324, Sibling 1000331, Sibling 1000343, Sibling 1000346, Sibling 1000348, Sibling 1000356, Sibling 1000357, Sibling 1000389, Sibling 1000405, Sibling 1000409, Sibling 1000414, Sibling 1000418, Sibling 1000425, Sibling 1000427, Sibling 1000451, Sibling 1000453, Sibling 1000458, Sibling 1000460, Sibling 1000463, Sibling 1000467, Sibling 1000473, Sibling 1000480, Sibling 1000481, Sibling 1000484, Sibling 1000487, Sibling 1000490, Sibling 1000492, Sibling 1000495, Sibling 1000496, Sibling 1000499, Sibling 1000514, Sibling 1000516, Sibling 1000517, Sibling 1000521, Sibling 1000523, Sibling 1000524, Sibling 1000527, Sibling 1000531, Sibling 1000534, Sibling 1000537, Sibling 1000541, Sibling 1000543, Sibling 1000547, Sibling 1000549, Sibling 1000551, Sibling 1000553, Sibling 1000557, Sibling 1000559, Sibling 1000561, Sibling 1000562, Sibling 1000567, Sibling 1000569, Sibling 1000571, Sibling 1000576, Sibling 1000582, Sibling 1000584, Sibling 1000588, Sibling 1000590, Sibling 1000598, Sibling 1000602, Sibling 1000604, Sibling 1000605, Sibling 1000611, Sibling 1000612, Sibling 1000615, Sibling 1000619, Sibling 1000622, Sibling 1000641, Sibling 1000644, Sibling 1000650, Sibling 1000652, Sibling 1000662, Sibling 1000669, Sibling 1000695, Sibling 1000706, Sibling 1000715, Sibling 1000716, Sibling 1000718, Sibling 1000723, Sibling 1000724, Sibling 1000725, Sibling 1000730, Sibling 1000732, Sibling 1000736, Sibling 1000740, Sibling 1000742, Sibling 1000761, Sibling 1000768, Sibling 1000770, Sibling 1000779, Sibling 1000780, Sibling 1000784, Sibling 1000809, Sibling 1000824, Sibling 1000826, Sibling 1000828, Sibling 1000830, Sibling 1000831, Sibling 1000835, Sibling 1000836, Sibling 1000840, Sibling 1000843, Sibling 1000846, Sibling 1000847, Sibling 1000849, Sibling 1000852, Sibling 1000856, Sibling 1000857, Sibling 1000858, Sibling 1000862, Sibling 1000864, Sibling 1000865, Sibling 1000869, Sibling 1000870, Sibling 1000872, Sibling 1000877, Sibling 1000880, Sibling 1000884, Sibling 1000886, Sibling 1000891, Sibling 1000892, Sibling 1000895, Sibling 1000897, Sibling 1000900, Sibling 1000906, Sibling 1000909, Sibling 1000916, Sibling 1000921, Sibling 1000924, Sibling 1000940, Sibling 1000946, Sibling 1000953, Sibling 1000955, Sibling 1000956, Sibling 1000993, Sibling 1001006, Sibling 1001009, Sibling 1001010, Sibling 1001013, Sibling 1001016, Sibling 1001019, Sibling 1001020, Sibling 1001036, Sibling 1001042, Sibling 1001048, Sibling 1001051, Sibling 1001056, Sibling 1001063, Sibling 1001066, Sibling 1001069, Sibling 1001070, Sibling 1001073, Sibling 1001091, Sibling 1001093, Sibling 1001108, Sibling 1001109, Sibling 1001113, Sibling 1001114, Sibling 1001116, Sibling 1001120, Sibling 1001123, Sibling 1001127, Sibling 1001128, Sibling 1001140, Sibling 1001142, Sibling 1001147, Sibling 1001148, Sibling 1001151, Sibling 1001155, Sibling 1001157, Sibling 1001159, Sibling 1001161, Sibling 1001165, Sibling 1001167, Sibling 1001169, Sibling 1001172, Sibling 1001176, Sibling 1001178, Sibling 1001188, Sibling 1001192, Sibling 1001205, Sibling 1001207, Sibling 1001208, Sibling 1001209, Sibling 1001211, Sibling 1001219, Sibling 1001221, Sibling 1001223, Sibling 1001231, Sibling 1001241, Sibling 1001253, Sibling 1001257, Sibling 1001295, Sibling 1001304, Sibling 1001317, Sibling 1001325, Sibling 1001333, Sibling 1001335, Sibling 1001337, Sibling 1001339, Sibling 1001343, Sibling 1001347, Sibling 1001349, Sibling 1001354, Sibling 1001356, Sibling 1001361, Sibling 1001366, Sibling 1001367, Sibling 1001370, Sibling 1001375, Sibling 1001378, Sibling 1001379, Sibling 1001382, Sibling 1001387, Sibling 1001388, Sibling 1001390, Sibling 1001395, Sibling 1001398, Sibling 1001400, Sibling 1001403, Sibling 1001409, Sibling 1001412, Sibling 1001433, Sibling 1001439, Sibling 1001441, Sibling 1001447, Sibling 1001449, Sibling 1001452, Sibling 1001456, Sibling 1001458, Sibling 1001460, Sibling 1001461, Sibling 1001462, Sibling 1001469, Sibling 1001472, Sibling 1001474, Sibling 1001477, Sibling 1001478, Sibling 1001481, Sibling 1001484, Sibling 1001488, Sibling 1001489, Sibling 1001491, Sibling 1001494, Sibling 1001496, Sibling 1001499, Sibling 1001503, Sibling 1001506, Sibling 1001509, Sibling 1001510, Sibling 1001527, Sibling 1001530, Sibling 1001533, Sibling 1001561, Sibling 1001565, Sibling 1001569, Sibling 1001571, Sibling 1001572, Sibling 1001575, Sibling 1001579, Sibling 1001581, Sibling 1001582, Sibling 1001586, Sibling 1001589, Sibling 1001590, Sibling 1001593, Sibling 1001597, Sibling 1001606, Sibling 1001610, Sibling 1001612, Sibling 1001624, Sibling 1001649, Sibling 1001651, Sibling 1001656, Sibling 1001660, Sibling 1001670, Sibling 1001681, Sibling 1001714, Sibling 1001721, Sibling 1001724, Sibling 1001727, Sibling 1001730, Sibling 1001731, Sibling 1001736, Sibling 1001744, Sibling 1001747, Sibling 1001748, Sibling 1001750, Sibling 1001752, Sibling 1001755, Sibling 1001762, Sibling 1001765, Sibling 1001767, Sibling 1001773, Sibling 1001776, Sibling 1001778, Sibling 1001781, Sibling 1001783, Sibling 1001784, Sibling 1001786, Sibling 1001788, Sibling 1001792, Sibling 1001794, Sibling 1001802, Sibling 1001805, Sibling 1001806, Sibling 1001811, Sibling 1001813, Sibling 1001817, Sibling 1001820, Sibling 1001823, Sibling 1001824, Sibling 1001829, Sibling 1001830, Sibling 1001835, Sibling 1001837, Sibling 1001841, Sibling 1001846, Sibling 1001848, Sibling 1001850, Sibling 1001852, Sibling 1001856, Sibling 1001857, Sibling 1001859, Sibling 1001875, Sibling 1001877, Sibling 1001880, Sibling 1001882, Sibling 1001884, Sibling 1001886, Sibling 1001895, Sibling 1001897, Sibling 1001898, Sibling 1001900, Sibling 1001905, Sibling 1001907, Sibling 1001912, Sibling 1001914, Sibling 1001916, Sibling 1001919, Sibling 1001921, Sibling 1001925, Sibling 1001928, Sibling 1001939, Sibling 1001952, Sibling 1001963, Sibling 1001981, Sibling 1001983, Sibling 1001985, Sibling 1001991, Sibling 1001995, Sibling 1001997, Sibling 1002000, Sibling 1002035, Sibling 1002039, Sibling 1002057, Sibling 1002067, Sibling 1002069, Sibling 1002071, Sibling 1002072, Sibling 1002074, Sibling 1002078, Sibling 1002082, Sibling 1002083, Sibling 1002103, Sibling 1002109, Sibling 1002111, Sibling 1002115, Sibling 1002119, Sibling 1002121, Sibling 1002123, Sibling 1002124, Sibling 1002128, Sibling 1002129, Sibling 1002131, Sibling 1002133, Sibling 1002136, Sibling 1002137, Sibling 1002139, Sibling 1002140, Sibling 1002142, Sibling 1002149, Sibling 1002153, Sibling 1002155, Sibling 1002159, Sibling 1002161, Sibling 1002164, Sibling 1002170, Sibling 1002174, Sibling 1002175, Sibling 1002180, Sibling 1002183, Sibling 1002191, Sibling 1002193, Sibling 1002196, Sibling 1002198, Sibling 1002200, Sibling 1002203, Sibling 1002205, Sibling 1002208, Sibling 1002210, Sibling 1002211, Sibling 1002221, Sibling 1002230, Sibling 1002236, Sibling 1002241, Sibling 1002243, Sibling 1002245, Sibling 1002247, Sibling 1002252, Sibling 1002256, Sibling 1002258, Sibling 1002259, Sibling 1002269, Sibling 1002273, Sibling 1002275, Sibling 1002279, Sibling 1002280, Sibling 1002285, Sibling 1002286, Sibling 1002292, Sibling 1002293, Sibling 1002295, Sibling 1002300, Sibling 1002301, Sibling 1002305, Sibling 1002309, Sibling 1002345, Sibling 1002349, Sibling 1002351, Sibling 1002371, Sibling 1002385, Sibling 1002391, Sibling 1002392, Sibling 1002397, Sibling 1002400, Sibling 1002406, Sibling 1002411, Sibling 1002414, Sibling 1002416, Sibling 1002420, Sibling 1002423, Sibling 1002425, Sibling 1002428, Sibling 1002433, Sibling 1002434, Sibling 1002445, Sibling 1002448, Sibling 1002450, Sibling 1002452, Sibling 1002454, Sibling 1002457, Sibling 1002460, Sibling 1002464, Sibling 1002468, Sibling 1002470, Sibling 1002473, Sibling 1002474, Sibling 1002477, Sibling 1002481, Sibling 1002483, Sibling 1002486, Sibling 1002489, Sibling 1002492, Sibling 1002493, Sibling 1002498, Sibling 1002509, Sibling 1002511, Sibling 1002517, Sibling 1002519, Sibling 1002520, Sibling 1002523, Sibling 1002531, Sibling 1002534, Sibling 1002538, Sibling 1002552, Sibling 1002555, Sibling 1002558, Sibling 1002566, Sibling 1002567, Sibling 1002569, Sibling 1002570, Sibling 1002575, Sibling 1002577, Sibling 1002579, Sibling 1002581, Sibling 1002583, Sibling 1002587, Sibling 1002589, Sibling 1002591, Sibling 1002594, Sibling 1002597, Sibling 1002599, Sibling 1002604, Sibling 1002607, Sibling 1002613, Sibling 1002614, Sibling 1002622, Sibling 1002624, Sibling 1002627, Sibling 1002629, Sibling 1002632, Sibling 1002635, Sibling 1002638, Sibling 1002644, Sibling 1002648, Sibling 1002651, Sibling 1002657, Sibling 1002659, Sibling 1002662, Sibling 1002677, Sibling 1002678, Sibling 1002679, Sibling 1002684, Sibling 1002687, Sibling 1002688, Sibling 1002690, Sibling 1002698, Sibling 1002700, Sibling 1002703, Sibling 1002706, Sibling 1002711, Sibling 1002714, Sibling 1002730, Sibling 1002732, Sibling 1002735, Sibling 1002745, Sibling 1002746, Sibling 1002752, Sibling 1002755, Sibling 1002760, Sibling 1002768, Sibling 1002779, Sibling 1002780, Sibling 1002784, Sibling 1002786, Sibling 1002792, Sibling 1002793, Sibling 1002795, Sibling 1002796, Sibling 1002800, Sibling 1002808, Sibling 1002810, Sibling 1002814, Sibling 1002815, Sibling 1002817, Sibling 1002822, Sibling 1002823, Sibling 1002827, Sibling 1002832, Sibling 1002839, Sibling 1002840, Sibling 1002844, Sibling 1002850, Sibling 1002852, Sibling 1002855, Sibling 1002856, Sibling 1002861, Sibling 1002864, Sibling 1002865, Sibling 1002870, Sibling 1002874, Sibling 1002876, Sibling 1002877, Sibling 1002882, Sibling 1002885, Sibling 1002889, Sibling 1002891, Sibling 1002893, Sibling 1002896, Sibling 1002904, Sibling 1002906, Sibling 1002907, Sibling 1002912, Sibling 1002914, Sibling 1002918, Sibling 1002920, Sibling 1002921, Sibling 1002924, Sibling 1002925, Sibling 1002931, Sibling 1002937, Sibling 1002939, Sibling 1002941, Sibling 1002944, Sibling 1002947, Sibling 1002953, Sibling 1002957, Sibling 1002963, Sibling 1002966, Sibling 1002969, Sibling 1002970, Sibling 1002973, Sibling 1002977, Sibling 1002982, Sibling 1002984, Sibling 1002986, Sibling 1002990, Sibling 1002994,

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### 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.