Truncated icosahedron

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 nonun


Duotone is a halftone reproduction of an image using the superimposition of one contrasting color halftone over another color halftone. This is most used to bring out middle tones and highlights of an image. Traditionally the superimposed contrasting halftone color is black and the most implemented colours are blue, yellow and red, however there are many varieties of color combinations used. Due to recent advances in technology, duotones and quadtones can be created using image manipulation programs. Duotone color mode in Adobe Photoshop computes the highlights and middle tones of a monochrome image in one color, allows the user to choose any color ink as the second color. A fake duotone, or duograph, is done by printing a single color with a one-color halftone over it; this process is not preferred over a regular duotone, as it loses much of the contrast of the image but it is easier and faster to create. Sepia tone Cyanotype Halftone Printmaking Color separation Luminous Landscape Duotone Article Article on B/W and toning Creative Pro Article Working with Duotone in Photoshop Duotone Photoshop Actions

Ales Prudnikau

Ales Prudnikau was a Belarusian poet. He was a cousin of Pavel Prudnikau. Ales Prudnikau was born into a peasant family, his father Traphim was called to the front at the time of World War I, he was badly injured and died soon after. Because of the unstable situation Ales couldn't find a constant place tor study for a long time. In 1924-1930 he was studying in the seven-years school in the village Miloslavicy. In 1930 he worked on the building of the railway Asipovichy - Mahilyow - Roslavl in 1931 he worked in Minsk. In 1932-1933 he was studying at the creative department of the Minsk pedagogical institute. In March 1933 he was arrested but after eight months was liberated and was called up to the Red Army, where he served until 1935. In 1935-1936 he worked on the local Klimovichy newspaper "Kamuna". In 1938 he graduated from the Leningrad pedagogical institute, he worked as the teacher as inspector of the Petrovo raion of Karelian ASSR, at the same time was studying at the philological faculty of the Karelian University.

Since the first days of the German invasion of the Soviet Union, he was an intelligence operator at the front. He was killed on August 1941 near the village Utuki of Kondopoga raion, he was rehabilitated in 1956. Ales Prudnikau began his creative activity as a school poet together with his cousin Pavel Prudnikau in 1926; the time of his studying coincided with the process of belarusization in BSSR, so it he began to write his verses in Belarusian. At the same time he was an informal correspondent for some newspapers. For example, after a treasure of ancient coins had been found in the village Stary Dzedzin, he wrote a report to the newspaper "Belarusian village" together with Pavel.. His first verses were published in 1930. In 1932 he wrote a poem "Stars of the Earth", published as a book where he told about buildings of the first five-year plan; some of his verses were published in the book "With the Blood of the Heart". Пруднікаў, А. Зямныя зоры / Алесь Пруднікаў. - Мн. Дзяржаўнае выдавецтва Беларусі, 1932.

Крывёю сэрца / укладальнік А. Вялюгін. - Мн. Беларусь, 1967. С. 116-122. Беларуская энцыклапедыя: У 18 т. Т. 13. - Мн.: БелЭн, 2001. - С. 48.Беларуская энцыклапедыя: У 18 т. Т. 13: Праміле - Рэлаксін/ Рэдкал.: Г. П. Пашкоў і інш. - Мн.: БелЭн, 2001. - 576 с.: іл. - С. 48. Беларускія пісьменнікі: Даведнік / Склад. А. К. Гардзінскі. А. Л. Верабей. - Мн. Маст. літ. 1994. - С. 441-442. Маракоў Л. Рэпрэсаваныя лiтаратары, навукоўцы, работнiкi асветы, грамадскiя i культурныя дзеячы Беларусі. 1794-1991. Том II. С. 159-160. Памяць: Гіст.-дакум. Хроніка Клімавіцк. Р-на. - Мн.: Універсітэцкае, 1995. - 645 с.: іл. - С. 624. Пруднікаў П. Далёкае, але не забытае: Успаміны. - Мн. Маст. літ. 1988. - 175 с. - С. 149-154. Рэпрэсаваныя лiтаратары, навукоўцы, работнiкi асветы, грамадскiя i культурныя дзеячы Беларусі. 1794-1991. Том II. ПРУДНІКАЎ Павел Іванавіч Алесь Пруднікаў. Біяграфія Pavel Prudnikau