Johannes Max Brückner was a German geometer, known for his collection of polyhedral models. Brückner was born on May 8, 1860 in Hartau, in the Kingdom of Saxony, a town, now part of Zittau, Germany, he completed a Ph. D. at Leipzig University in 1886, supervised by Felix Klein and Wilhelm Scheibner, with a dissertation concerning conformal maps. After teaching at a grammar school in Zittau, he moved to the gymnasium in Bautzen. Brückner is known for making many geometric models of stellated and uniform polyhedra, which he documented in his book Vielecke und Vielflache: Theorie und Geschichte; the shapes first studied in this book include the final stellation of the icosahedron and the compound of three octahedra, made famous by M. C. Escher's print Stars. Joseph Malkevitch lists the publication of this book, which documented all, known on polyhedra at the time, as one of 25 milestones in the history of polyhedra. Malkevitch writes that the book's "beautiful pictures of uniform polyhedra... served as an inspiration to people later".
Brückner was an invited speaker at the International Congress of Mathematicians in 1904, 1908, 1912, 1928. In 1930–1931 he donated his model collection to Heidelberg University, the university in turn gave him an honorary doctorate in 1931. Brückner died on November 1, 1934 in Bautzen
A polyhedron model is a physical construction of a polyhedron, constructed from cardboard, plastic board, wood board or other panel material, or, less solid material. Since there are 75 uniform polyhedra, including the five regular convex polyhedra, five polyhedral compounds, four Kepler-Poinsot polyhedra, thirteen Archimedean solids, constructing or collecting polyhedron models has become a common mathematical recreation. Polyhedron models are found in mathematics classrooms much as globes in geography classrooms. Polyhedron models are notable as three-dimensional proof-of-concepts of geometric theories; some polyhedra make great centerpieces, tree toppers, Holiday decorations, or symbols. The Merkaba religious symbol, for example, is a stellated octahedron. Constructing large models offer challenges in engineering structural design. Construction begins by choosing a size of the model, either the length of its edges or the height of the model; the size will dictate the material, the adhesive for edges, the construction time and the method of construction.
The second decision involves colours. A single-colour cardboard model is easiest to construct — and some models can be made by folding a pattern, called a net, from a single sheet of cardboard. Choosing colours requires geometric understanding of the polyhedron. One way is to colour each face differently. A second way is to colour all square faces the same, all pentagonal faces the same, so forth. A third way is to colour opposite faces the same. Many polyhedra are coloured such that no same-coloured faces touch each other along an edge or at a vertex. For example, a 20-face icosahedron can use twenty colours, one colour, ten colours, or five colours, respectively. An alternative way for polyhedral compound models is to use a different colour for each polyhedron component. Net templates are made. One way is to copy templates from a polyhedron-making book, such as Magnus Wenninger's Polyhedron Models, 1974. A second way is drawing faces on paper or with computer-aided design software and drawing on them the polyhedron's edges.
The exposed nets of the faces are traced or printed on template material. A third way is using the software named Stella to print nets. A model a large one, may require another polyhedron as its inner structure or as a construction mold. A suitable inner structure prevents the model from collapsing from stress; the net templates are replicated onto the material, matching the chosen colours. Cardboard nets are cut with tabs on each edge, so the next step for cardboard nets is to score each fold with a knife. Panelboard nets, on the other hand, require molds and cement adhesives. Assembling multi-colour models is easier with a model of a simpler related polyhedron used as a colour guide. Complex models, such as stellations, can have hundreds of polygons in their nets. Recent computer graphics technologies allow people to rotate 3D polyhedron models on a computer video screen in all three dimensions. Recent technologies provide shadows and textures for a more realistic effect. Polyhedron List of Wenninger polyhedron models Stella: Polyhedron Navigator: Software to explore virtual polyhedra and print their nets to enable physical construction Interactive 3D polyhedra in Java Wooden Polyhedra Models George Hart's extensive encyclopedia of polyhedra George Hart's Pavilion of Polyhedreality Online rotatable polyhedron models WOODEN POLYHEDRA 30
Mammals are vertebrate animals constituting the class Mammalia, characterized by the presence of mammary glands which in females produce milk for feeding their young, a neocortex, fur or hair, three middle ear bones. These characteristics distinguish them from reptiles and birds, from which they diverged in the late Triassic, 201–227 million years ago. There are around 5,450 species of mammals; the largest orders are the rodents and Soricomorpha. The next three are the Primates, the Cetartiodactyla, the Carnivora. In cladistics, which reflect evolution, mammals are classified as endothermic amniotes, they are the only living Synapsida. The early synapsid mammalian ancestors were sphenacodont pelycosaurs, a group that produced the non-mammalian Dimetrodon. At the end of the Carboniferous period around 300 million years ago, this group diverged from the sauropsid line that led to today's reptiles and birds; the line following the stem group Sphenacodontia split off several diverse groups of non-mammalian synapsids—sometimes referred to as mammal-like reptiles—before giving rise to the proto-mammals in the early Mesozoic era.
The modern mammalian orders arose in the Paleogene and Neogene periods of the Cenozoic era, after the extinction of non-avian dinosaurs, have been among the dominant terrestrial animal groups from 66 million years ago to the present. The basic body type is quadruped, most mammals use their four extremities for terrestrial locomotion. Mammals range in size from the 30–40 mm bumblebee bat to the 30-meter blue whale—the largest animal on the planet. Maximum lifespan varies from two years for the shrew to 211 years for the bowhead whale. All modern mammals give birth to live young, except the five species of monotremes, which are egg-laying mammals; the most species-rich group of mammals, the cohort called placentals, have a placenta, which enables the feeding of the fetus during gestation. Most mammals are intelligent, with some possessing large brains, self-awareness, tool use. Mammals can communicate and vocalize in several different ways, including the production of ultrasound, scent-marking, alarm signals and echolocation.
Mammals can organize themselves into fission-fusion societies and hierarchies—but can be solitary and territorial. Most mammals are polygynous. Domestication of many types of mammals by humans played a major role in the Neolithic revolution, resulted in farming replacing hunting and gathering as the primary source of food for humans; this led to a major restructuring of human societies from nomadic to sedentary, with more co-operation among larger and larger groups, the development of the first civilizations. Domesticated mammals provided, continue to provide, power for transport and agriculture, as well as food and leather. Mammals are hunted and raced for sport, are used as model organisms in science. Mammals have been depicted in art since Palaeolithic times, appear in literature, film and religion. Decline in numbers and extinction of many mammals is driven by human poaching and habitat destruction deforestation. Mammal classification has been through several iterations since Carl Linnaeus defined the class.
No classification system is universally accepted. George Gaylord Simpson's "Principles of Classification and a Classification of Mammals" provides systematics of mammal origins and relationships that were universally taught until the end of the 20th century. Since Simpson's classification, the paleontological record has been recalibrated, the intervening years have seen much debate and progress concerning the theoretical underpinnings of systematization itself through the new concept of cladistics. Though field work made Simpson's classification outdated, it remains the closest thing to an official classification of mammals. Most mammals, including the six most species-rich orders, belong to the placental group; the three largest orders in numbers of species are Rodentia: mice, porcupines, beavers and other gnawing mammals. The next three biggest orders, depending on the biological classification scheme used, are the Primates including the apes and lemurs. According to Mammal Species of the World, 5,416 species were identified in 2006.
These were grouped into 153 families and 29 orders. In 2008, the International Union for Conservation of Nature completed a five-year Global Mammal Assessment for its IUCN Red List, which counted 5,488 species. According to a research published in the Journal of Mammalogy in 2018, the number of recognized mammal species is 6,495 species included 96 extinct; the word "mammal" is modern, from the scientific name Mammalia coined by Carl Linnaeus in 1758, derived from the Latin mamma. In an influential 1988 paper, Timothy Rowe defined Mammalia phylogenetically as the crown group of mammals, the clade consisting of the most recent common ancestor of living monotremes and therian m
In geometry, the truncated icosahedron is an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose faces are two or more types of regular polygons. It has 20 regular hexagonal faces, 60 vertices and 90 edges, it is 1,1, containing pentagonal and hexagonal faces. This geometry is associated with footballs patterned with white hexagons and black pentagons. Geodesic domes such as those whose architecture Buckminster Fuller pioneered are based on this structure, it corresponds to the geometry of the fullerene C60 molecule. It is used in the cell-transitive hyperbolic space-filling tessellation, the bitruncated order-5 dodecahedral honeycomb; this polyhedron can be constructed from an icosahedron with the 12 vertices truncated such that one third of each edge is cut off at each of both ends. This creates 12 new pentagon faces, leaves the original 20 triangle faces as regular hexagons, thus the length of the edges is one third of that of the original edges. Cartesian coordinates for the vertices of a truncated icosahedron centered at the origin are all permutations of: where φ = 1 + √5/2 is the golden mean.
The circumradius is √9φ + 10 ≈ 4.956 and the edges have length 2. The truncated icosahedron has five special orthogonal projections, centered, on a vertex, on two types of edges, two types of faces: hexagonal and pentagonal; the last two correspond to the A2 and H2 Coxeter planes. The truncated icosahedron can be represented as a spherical tiling, projected onto the plane via a stereographic projection; this projection is conformal, preserving angles but not lengths. Straight lines on the sphere are projected as circular arcs on the plane. If the edge length of a truncated icosahedron is a, the radius of a circumscribed sphere is: r u = a 2 1 + 9 φ 2 = a 4 58 + 18 5 ≈ 2.478 018 66 a where φ is the golden ratio. This result is easy to get by using one of the three orthogonal golden rectangles drawn into the original icosahedron as the starting point for our considerations; the angle between the segments joining the center and the vertices connected by shared edge is 23.281446°. The area A and the volume V of the truncated icosahedron of edge length a are: A = a 2 ≈ 72.607 253 a 2 V = 125 + 43 5 4 a 3 ≈ 55.287 7308 a 3.
With unit edges, the surface area is 21 for the pentagons and 52 for the hexagons, together 73. The truncated icosahedron demonstrates the Euler characteristic: 32 + 60 − 90 = 2; the balls used in association football and team handball are the best-known example of a spherical polyhedron analog to the truncated icosahedron, found in everyday life. The ball comprises the same pattern of regular pentagons and regular hexagons, but it is more spherical due to the pressure of the air inside and the elasticity of the ball; this ball type was introduced to the World Cup in 1970. Geodesic domes are based on triangular facetings of this geometry with example structures found across the world, popularized by Buckminster Fuller. A variation of the icosahedron was used as the basis of the honeycomb wheels used by the Pontiac Motor Division between 1971 and 1976 on its Trans Am and Grand Prix; this shape was the configuration of the lenses used for focusing the explosive shock waves of the detonators in both the gadget and Fat Man atomic bombs.
The truncated icosahedron can be described as a model of the Buckminsterfullerene, or "buckyball," molecule, an allotrope of elemental carbon, discovered in 1985. The diameter of the football and the fullerene molecule are 22 cm and about 0.71 nm hence the size ratio is ≈31,000,000:1. In popular craft culture, large sparkleballs can be made using a icosahedron pattern and plastic, styrofoam or paper cups. A truncated icosahedron with "solid edges" by Leonardo da Vinci appears as an illustration in Luca Pacioli's book De divina proportione; these uniform star-polyhedra, one icosahedral stellation have nonuniform truncated icosa
Father Magnus J. Wenninger OSB was an American mathematician who worked on constructing polyhedron models, wrote the first book on their construction. Born to German immigrants in Park Falls, Joseph Wenninger always knew he was going to be a priest. From an early age, it was understood that his brother Heinie would take after their father and become a baker, that Joe, as he was known, would go into the priesthood; when Wenninger was thirteen, after graduating from the parochial school in Park Falls, his parents saw an advertisement in the German newspaper Der Wanderer that would help to shape the rest of his life. The ad was for a preparatory school in Collegeville, associated with the Benedictine St. John’s University. While admitting to feeling homesick at first, Wenninger made friends and, after a year, knew that this was where he needed to be, he was a student in a section of the prep school that functioned as a "minor seminary" – moving on into St. John’s where he studied philosophy and theology, which led into the priesthood.
When Fr. Wenninger became a Benedictine monk, he took on his monastic name Magnus, meaning "Great". At the start of his career, Wenninger did not set out on a path one might expect would lead to his becoming the great polyhedronist that he is known as today. Rather, a few chance happenings and minor decisions shaped a course for Wenninger that led to his groundbreaking studies. Shortly after becoming a priest, Wenninger’s Abbot informed him that their order was starting up a school in the Bahamas, it was decided. In order to do this, it was necessary. Wenninger was sent in Canada, to study educational psychology. There he studied symbolic logic under Thomas Greenwood of the philosophy department, his thesis title was "The Concept of Number According to Roger Bacon and Albert the Great". After completing his degree, Wenninger went to the school in the Bahamas, where he was asked by the headmaster to choose between teaching English or math. Wenninger chose math. However, not having taken many math courses in college, Wenninger admits to being able to teach by staying a few pages ahead of the students.
He taught Algebra, Euclidean Geometry and Analytic Geometry. After ten years of teaching, Wenninger felt. At the suggestion of his headmaster, Wenninger attended the Columbia Teachers College in summer sessions over a four-year period in the late fifties, it was here that his studies of the polyhedra began. Wenninger died at the age of 97, at St John's Abbey on Friday, February 17, 2017. Wenninger’s first publication on the topic of polyhedra was the booklet entitled, "Polyhedron Models for the Classroom", which he wrote in 1966, he wrote to H. S. M. Coxeter and received a copy of Uniform polyhedra which had a complete list of all 75 uniform polyhedra. After this, he spent a great deal of time building various polyhedra, he had them on display in his classroom. At this point, Wenninger decided to contact a publisher to see, he had the models photographed and wrote the accompanying text, which he sent off to Cambridge University Press in London. The publishers indicated an interest in the book only if Wenninger built all 75 of the uniform polyhedra.
Wenninger did complete the models, with the help of R. Buckley of Oxford University who had done the calculations for the snub forms by computer; this allowed Wenninger to build these difficult polyhedra with the exact measurements for lengths of the edges and shapes of the faces. This was the first time; this project took Wenninger nearly ten years, the book, Polyhedron Models, was published by the Cambridge University Press in 1971 due to the exceptional photographs taken locally in Nassau. From 1971 onward, Wenninger focused his attention on the projection of the uniform polyhedra onto the surface of their circumscribing spheres; this led to the publication of his second book, Spherical Models in 1979, showing how regular, or semiregular, polyhedron can be used to build geodesic domes. He exchanged ideas with other mathematicians, Hugo Verheyen and Gilbert Fleurent. In 1981, Wenninger returned to St. John's Abbey, his third book, Dual Models, appeared in 1983. The book is a sequel to Polyhedron Models, since it includes instructions on how to make paper models of the duals of all 75 uniform polyhedra.
List of Wenninger polyhedron models Banchoff, Father Magnus and his polyhedrons, LAB Issue 02, June 2008 Friedman, Nat. "Magnus Wenninger: Mathematical Models", Hyperseeing Wenninger, Polyhedron Models, Cambridge University Press, ISBN 978-0-521-09859-5, MR 0467493 Wenninger, Spherical Models, Cambridge University Press, ISBN 978-0-521-29432-4, MR 0552023, archived from the original on July 4, 2008 Reprinted by Dover 1999 ISBN 978-0-486-40921-4 Wenninger, Dual Models, Cambridge University Press, doi:10.1017/CBO9780511569371, ISBN 978-0-521-54325-5, MR 0730208Complete publications: 1963-69 Stellated Rhombic Dodecahedron Puzzle The Mathematics Teacher. The World of Polyhedrons The Mathematics Teacher; some Facts About Uniform Polyhedrons. Summation: Association of Teachers of Mathematics of New York City. 11:6 33-35. Fancy Shapes from Geometric Figures. Grade Teacher 84:4 61-63, 129-130. 1970-79 Polyhedron Models for the Classroom National Coun
Small stellated dodecahedron
In geometry, the small stellated dodecahedron is a Kepler-Poinsot polyhedron, named by Arthur Cayley, with Schläfli symbol. It is one of four nonconvex regular polyhedra, it is composed with five pentagrams meeting at each vertex. It shares the same vertex arrangement as the convex regular icosahedron, it shares the same edge arrangement with the great icosahedron, with which it forms a degenerate uniform compound figure. It is the second of four stellations of the dodecahedron; the small stellated dodecahedron can be constructed analogously to the pentagram, its two-dimensional analogue, via the extension of the edges of the core polytope until a point is reached where they intersect. If the pentagrammic faces are considered as 5 triangular faces, it shares the same surface topology as the pentakis dodecahedron, but with much taller isosceles triangle faces, with the height of the pentagonal pyramids adjusted so that the five triangles in the pentagram become coplanar; the critical angle is atan above the dodecahedron face.
If we regard it as having 12 pentagrams as faces, with these pentagrams meeting at 30 edges and 12 vertices, we can compute its genus using Euler's formula V − E + F = 2 − 2 g and conclude that the small stellated dodecahedron has genus 4. This observation, made by Louis Poinsot, was confusing, but Felix Klein showed in 1877 that the small stellated dodecahedron could be seen as a branched covering of the Riemann sphere by a Riemann surface of genus 4, with branch points at the center of each pentagram. In fact this Riemann surface, called Bring's curve, has the greatest number of symmetries of any Riemann surface of genus 4: the symmetric group S 5 acts as automorphisms It can be seen in a floor mosaic in St Mark's Basilica, Venice by Paolo Uccello circa 1430, it is central to two lithographs by M. C. Escher: Contrast and Gravitation, its convex hull is the regular convex icosahedron. It shares its edges with the great icosahedron. There are four related uniform polyhedra, constructed as degrees of truncation.
The dual is a great dodecahedron. The dodecadodecahedron is a rectification; the truncated small stellated dodecahedron can be considered a degenerate uniform polyhedron since edges and vertices coincide, but it is included for completeness. Visually, it looks like a regular dodecahedron on the surface, but it has 24 faces in overlapping pairs; the spikes are truncated. The 24 faces are 12 pentagons from the truncated vertices and 12 decagons taking the form of doubly-wound pentagons overlapping the first 12 pentagons; the latter faces are formed by truncating the original pentagrams. When an -gon is truncated, it becomes a -gon. For example, a truncated pentagon becomes a decagon, so truncating a pentagram becomes a doubly-wound pentagon. Compound of small stellated dodecahedron and great dodecahedron Wenninger, Magnus. Polyhedron Models. Cambridge University Press. ISBN 0-521-09859-9. Weber, Matthias, "Kepler's small stellated dodecahedron as a Riemann surface", Pacific J. Math. 220: 167–182, doi:10.2140/pjm.2005.220.167 Eric W. Weisstein, Small stellated dodecahedron at MathWorld.
Weisstein, Eric W. "DodecahedronStellations". MathWorld. Uniform polyhedra and duals