## 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-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.

### 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:RammböckeOldenburg, Lucky Number Slevin, BaumBanane, Holgor, Fabian'scher Rupertieder, Amy, Polylop, Tony, Cutie, Klunki von Glitzenstein, EinSüßesPolly, Kimongo mit der Armstütze, Majula, ALINEDER, Amhehselaner, coolster Pudding, schon besetzt (Zustand), Elli, Peyer, buscho, . . .
Birgit, Jonas, Josis schiefer Regenschirm, luca-franz, Totokaeder, Bitto, Polytraum , Sternchen im Himmel, F.F.E, Hai!, LIA, Merida, Rainbow-Spitze, Thomas, Mathe, Candy, Bunter Papagei, Shawn, Dave Thomas, minus, Mrs. Floppeck, Gachibus, Emiliaeder.10 ❤️, Audrey, Sturnus, Flacheck, Luca Schreiner, MoeyFilou, Flamingo03, Alfred, Hank, Biene, Die Schulzahl P-Siegurd, Lisa, Petronio, Dentheda, Marc-Uwe Pi, Maximilian , Spikey McSpikeFace, Spitzchen, Diamantik, Olivia, Jakiki6, lici, Otto Klaus, Poly, A wild thing, pingpong, Anna, Polytopianna, Kamil, Rüdigeder, Cardi B, Sophias cooler Polyeder , Kattaeder, Max, Hatschibu, Felix 123, Festucam, extragross, Tirolpol, Diamand ohne feinschliff, Polyti, Tresa, Der Schredder, Rehmani, Patrick, Korpilopia, Claramariaeder, Frontmann Aste, JuliaSchmiedel, Stefan aus Halle, PolyGaR, Capyhedron, Egon, Annchen, Polydue :-) :D , Pfeilspitze, Der schönste Diamant, Mellanay, Zelleragon, Polykatzus, Silica, Bigmac-Eder, Ricarda Maria Kemper, Polster, Anna-Polyeder, Hugo der Hässliche, mimimimu, Pia, Melusine, Black Diademe, /87, Mimi_22, Lord Master, Gymnasium H2O PAUL G, Corvolo, Erlkönig, Polycedric, Kevin, Hoxlpoxl, Flips, Harry, Nele, Olilini, Kerstin, Andrei, Fluffy, Triangle, Leibigeder, Jan Urbas, Pollyeder, Oliver, Naya, Ungeschickt, Pikachu, Dodoeder , Drumwickelst, Kantenberg, Error404, Ella, Kyros der Große , Polyna, Mme Polly, TobiasAlexanderJürgenFörster, Puddle , Piquah, Lissi, Mowieder, Keksuel, Krabine, Ronny, Nineteder, Phillip-Polyeder, Paradox, Mankebert, Eckard, arnold, Bionik, Mr. Malla, tofu, Mashyeder, Fanny Hensel, Schery, Paulaeder, Fredoeder, Tintin , Fuchsmützchen, Rando Jr., Luisali, Jens, Bastianeder, MatheAdvebt, Brabbilinchen, Eva Silverstein, Gabrielegon, Marina, Löwe, Felix, Mr.Eisbär, Davemuhelogon, HighVoltageRachel, DETR0MEE11, Baumhauspolyeder, Hanna, Wachihedra, Mihops, Alexeder, TRUform, Flash, Bärchen, 90 Waterdell Lane, Adoeder, Aimooser, Sokath, Tammertop, Lissy von RL, Boris, tobert, Anduriel, Antonia, Poly-Eddy, Polly der Polyeder, Polykampsches Witteck, Quantoeder, Joanna, Charlotte Bühler, James, Hacke-P-Eder, Pogidoodle, polypotatoe, Tigal, Der_Maniac, Theo, Cristalo, Necesitogasolina, Polyelster, Polyphilipp, Klaus-Erwin Ankerplatz1, Kurt, Yayayetoptopjambo, Alexandra , Ed, Tomo, Helgaedron, Miro, Epolyetischopeder, Halogenid, Karsten2.0, mowaii, Eispokrilystall, Stefan110377, Sonnenschein, oumuamua, Gert, Keilbert, LUCKY!, Sophie et Félix arbeiten, Lazareder, Tipped quasi-cone, Nora, Polymuckl, Dhana‘s Poly, Finn, Pointyhedron, 901206, haariger iggi, Poly-Laufen, Knurps, Tierchen, eleluk, Pflaumeh, Leoaeder, PolyDachs, Greenhorn2, Remio, Mehreck, Frieda, Tasse, Tolles Familienzelt, H-Milach, diamnt, Lenhatu_1996, Ursula, MyNameIs Jeff, Leatop, Kurti, Daduda, Polly , Herr Biernot, Polytoby, Alaskanadalaskalachs, Addison, lul, Sergio, Biegemann, Jimmy, 1/2 & Ganze Wahrheit Reimunds, Käsestück, Muddopia, Wuppi, Pauly, April, Frida123, Sistaaa-eder, Ronnyeder, eins Lemurenpack, Jane, FEDERICOPAPAGEI, agent general , Marco Polo, Féileacán, 10 - SEITI, Dixieder, Chrijoma, Meister Polyeder, Julius, Lillipop, Experimentier-Werkstatt, jillometer, Puffel, alin, Matthias-Eder, Shirkan, Junge Junge, Eckchen, ARVM, Snufkin’s Snout, Gloria Borger, Lexa, Besig Bitch , Gammelchen, FrEDERik, Mona, Gregyeder, Lost In Space, TähtiGwiazdaStjärnaЗвез, Smilodon, Fraíche_Poly, Caro Straub, Jin, Wolfseck, Freddy, Emitinmar, Leilani, Peiki, Dachs, Eddy von Polyeder, Matthis_Polydragon, Sibling 900015, Sibling 900029, Sibling 900033, Sibling 900037, Sibling 900040, Sibling 900068, Sibling 900077, Sibling 900081, Sibling 900083, Sibling 900091, Sibling 900095, Sibling 900100, Sibling 900115, Sibling 900129, Sibling 900156, Sibling 900168, Sibling 900176, Sibling 900185, Sibling 900193, Sibling 900197, Sibling 900200, Sibling 900202, Sibling 900206, Sibling 900219, Sibling 900227, Sibling 900239, Sibling 900246, Sibling 900262, Sibling 900271, Sibling 900286, Sibling 900292, Sibling 900294, Sibling 900295, Sibling 900302, Sibling 900313, Sibling 900331, Sibling 900334, Sibling 900338, Sibling 900341, Sibling 900343, Sibling 900351, Sibling 900368, Sibling 900375, Sibling 900379, Sibling 900382, Sibling 900401, Sibling 900405, Sibling 900411, Sibling 900418, Sibling 900421, Sibling 900430, Sibling 900437, Sibling 900441, Sibling 900449, Sibling 900451, Sibling 900453, Sibling 900470, Sibling 900480, Sibling 900484, Sibling 900490, Sibling 900493, Sibling 900501, Sibling 900513, Sibling 900514, Sibling 900517, Sibling 900522, Sibling 900529, Sibling 900541, Sibling 900550, Sibling 900562, Sibling 900565, Sibling 900601, Sibling 900614, Sibling 900618, Sibling 900620, Sibling 900622, Sibling 900639, Sibling 900642, Sibling 900644, Sibling 900645, Sibling 900648, Sibling 900675, Sibling 900679, Sibling 900688, Sibling 900691, Sibling 900701, Sibling 900704, Sibling 900705, Sibling 900717, Sibling 900719, Sibling 900720, Sibling 900727, Sibling 900730, Sibling 900733, Sibling 900735, Sibling 900753, Sibling 900777, Sibling 900799, Sibling 900817, Sibling 900819, Sibling 900824, Sibling 900826, Sibling 900833, Sibling 900835, Sibling 900837, Sibling 900849, Sibling 900852, Sibling 900856, Sibling 900868, Sibling 900873, Sibling 900877, Sibling 900879, Sibling 900897, Sibling 900901, Sibling 900911, Sibling 900916, Sibling 900924, Sibling 900927, Sibling 900938, Sibling 900942, Sibling 900949, Sibling 900956, Sibling 900959, Sibling 900992, Sibling 900993, Sibling 900996, Sibling 900998, Sibling 901000, Sibling 901003, Sibling 901009, Sibling 901029, Sibling 901066, Sibling 901070, Sibling 901071, Sibling 901081, Sibling 901091, Sibling 901103, Sibling 901104, Sibling 901108, Sibling 901113, Sibling 901172, Sibling 901178, Sibling 901183, Sibling 901187, Sibling 901209, Sibling 901212, Sibling 901214, Sibling 901218, Sibling 901221, Sibling 901226, Sibling 901227, Sibling 901228, Sibling 901231, Sibling 901233, Sibling 901265, Sibling 901269, Sibling 901272, Sibling 901275, Sibling 901305, Sibling 901311, Sibling 901319, Sibling 901325, Sibling 901339, Sibling 901356, Sibling 901372, Sibling 901373, Sibling 901397, Sibling 901401, Sibling 901404, Sibling 901422, Sibling 901424, Sibling 901425, Sibling 901427, Sibling 901455, Sibling 901457, Sibling 901460, Sibling 901461, Sibling 901467, Sibling 901474, Sibling 901475, Sibling 901483, Sibling 901493, Sibling 901511, Sibling 901523, Sibling 901530, Sibling 901552, Sibling 901555, Sibling 901559, Sibling 901561, Sibling 901575, Sibling 901583, Sibling 901587, Sibling 901589, Sibling 901591, Sibling 901596, Sibling 901598, Sibling 901604, Sibling 901609, Sibling 901620, Sibling 901628, Sibling 901632, Sibling 901634, Sibling 901641, Sibling 901652, Sibling 901663, Sibling 901668, Sibling 901675, Sibling 901685, Sibling 901688, Sibling 901690, Sibling 901696, Sibling 901706, Sibling 901708, Sibling 901718, Sibling 901727, Sibling 901743, Sibling 901751, Sibling 901756, Sibling 901757, Sibling 901759, Sibling 901762, Sibling 901769, Sibling 901771, Sibling 901777, Sibling 901784, Sibling 901790, Sibling 901802, Sibling 901809, Sibling 901811, Sibling 901813, Sibling 901814, Sibling 901823, Sibling 901831, Sibling 901834, Sibling 901835, Sibling 901839, Sibling 901840, Sibling 901843, Sibling 901853, Sibling 901856, Sibling 901858, Sibling 901887, Sibling 901891, Sibling 901895, Sibling 901902, Sibling 901911, Sibling 901918, Sibling 901937, Sibling 901943, Sibling 901952, Sibling 901957, Sibling 902001, Sibling 902004, Sibling 902012, Sibling 902042, Sibling 902049, Sibling 902050, Sibling 902053, Sibling 902057, Sibling 902076, Sibling 902084, Sibling 902130, Sibling 902158, Sibling 902161, Sibling 902171, Sibling 902174, Sibling 902184, Sibling 902187, Sibling 902204, Sibling 902254, Sibling 902259, Sibling 902271, Sibling 902278, Sibling 902280, Sibling 902289, Sibling 902297, Sibling 902311, Sibling 902313, Sibling 902335, Sibling 902374, Sibling 902395, Sibling 902405, Sibling 902407, Sibling 902409, Sibling 902415, Sibling 902418, Sibling 902423, Sibling 902428, Sibling 902431, Sibling 902439, Sibling 902457, Sibling 902485, Sibling 902487, Sibling 902503, Sibling 902509, Sibling 902218, Sibling 902222, Sibling 902225, Sibling 902228, Sibling 902234, Sibling 902242, Sibling 902245,

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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.