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Polycubes
upAll 8 one-sided tetracubes – if chirality is ignored, the bottom 2 in grey are considered the same, giving 7 free tetracubes in total A puzzle involving arranging nine L tricubes into a 3×3×3 cube A polycube is a solid figure formed by joining one or more equal cubes face to face. Polycubes are the three-dimensional analogues of the planar polyominoes. The Soma cube, the Bedlam cube, the Diabolical cube, the Slothouber–Graatsma puzzle, and the Conway puzzle are examples of packing problems based on polycubes. Enumerating polycubes A chiral pentacube Like polyominoes, polycubes can be enumerated in two ways, depending on whether chiral pairs of polycubes (those equivalent by mirror reflection, but not by using only translations and rotations) are counted as one polycube or two. For example, 6 tetracubes are achiral and one is chiral, giving a count of 7 or 8 tetracubes respectively. Unlike polyominoes, polycubes are usually counted with mirror pairs distinguished, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tetracube Categories
image:tetracube_categories.svg, upAll 8 one-sided tetracubes – if chirality is ignored, the bottom 2 in grey are considered the same, giving 7 free tetracubes in total image:9L cube puzzle solution.svg, A puzzle involving arranging nine L tricubes into a 3×3×3 cube A polycube is a solid figure formed by joining one or more equal cube (geometry), cubes face to face. Polycubes are the three-dimensional analogues of the planar polyominoes. The Soma cube, the Bedlam cube, the Diabolical cube, the Slothouber–Graatsma puzzle, and the Conway puzzle are examples of packing problems based on polycubes. Enumerating polycubes image:AGK-pentacube.png, A Chirality (mathematics), chiral pentacube Like polyominoes, polycubes can be enumerated in two ways, depending on whether Chirality (mathematics), chiral pairs of polycubes (those equivalent by Reflection symmetry, mirror reflection, but not by using only translations and rotations) are counted as one polycube or two. For example, 6 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diabolical Cube
The diabolical cube is a three-dimensional dissection puzzle consisting of six polycubes (shapes formed by gluing cubes together face to face) that can be assembled together to form a single 3 × 3 × 3 cube.. The six pieces are: one dicube, one tricube, one tetracube, one pentacube, one hexacube and one heptacube, that is, polycubes of 2, 3, 4, 5, 6 and 7 cubes. There are many similar variations of this type of puzzle, including the Soma cube The Soma cube is a mechanical puzzle#Assembly, solid dissection puzzle invented by Danish polymath Piet Hein (scientist), Piet Hein in 1933 during a lecture on quantum mechanics conducted by Werner Heisenberg. Seven different Polycube, pieces ... and the Slothouber–Graatsma puzzle, two other dissections of a 3 × 3 × 3 cube into polycubes which use seven and nine pieces respectively. However, writes that the diabolical cube appears to be the oldest puzzle of this type, first appearing in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polyomino
A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge. It is a polyform whose cells are squares. It may be regarded as a finite subset of the regular square tiling. Polyominoes have been used in popular puzzles since at least 1907, and the enumeration of pentominoes is dated to antiquity. Many results with the pieces of 1 to 6 squares were first published in '' Fairy Chess Review'' between the years 1937 and 1957, under the name of "dissection problems." The name ''polyomino'' was invented by Solomon W. Golomb in 1953, and it was popularized by Martin Gardner in a November 1960 " Mathematical Games" column in ''Scientific American''. Related to polyominoes are polyiamonds, formed from equilateral triangles; polyhexes, formed from regular hexagons; and other plane polyforms. Polyominoes have been generalized to higher dimensions by joining cubes to form polycubes, or hypercubes to form polyhypercubes. In statistical physics, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Soma Cube
The Soma cube is a mechanical puzzle#Assembly, solid dissection puzzle invented by Danish polymath Piet Hein (scientist), Piet Hein in 1933 during a lecture on quantum mechanics conducted by Werner Heisenberg. Seven different Polycube, pieces made out of unit cubes must be assembled into a 3×3×3 cube. The pieces can also be used to make a variety of other Three-dimensional space, 3D shapes. The pieces of the Soma cube consist of all possible combinations of at most four unit cubes, joined at their faces, such that at least one inside corner is formed. There are no combinations of one or two cubes that satisfy this condition, but one combination of three cubes and six combinations of four cubes that do. Thus, 3 + (6 × 4) is 27, which is exactly the number of cells in a 3×3×3 cube. Of these seven combinations, two are mirror images of each other (see Chirality (mathematics), Chirality). The Soma cube was popularized by Martin Gardner in the September 1958 Mathematical Games ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Slothouber–Graatsma Puzzle
The Slothouber–Graatsma puzzle is a packing problem that calls for packing six 1 × 2 × 2 blocks and three 1 × 1 × 1 blocks into a 3 × 3 × 3 box. The solution to this puzzle is unique (up to mirror reflections and rotations). It was named after its inventors Jan Slothouber and William Graatsma. The puzzle is essentially the same if the three 1 × 1 × 1 blocks are left out, so that the task is to pack six 1 × 2 × 2 blocks into a cubic box with volume 27. Solution The solution of the Slothouber–Graatsma puzzle is straightforward when one realizes that the three 1 × 1 × 1 blocks (or the three holes) need to be placed along a body diagonal of the box, as each of the 3 x 3 layers in the various directions needs to contain such a unit block. This follows from parity (mathematics), parity considerations, because the larger blocks can only fill an even number of the 9 cells in each 3 x 3 layer.Elwyn R. Berlekamp, John H. Conway and Richard K. Guy: Winning ways for your math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crucifixion (Corpus Hypercubus)
''Crucifixion (Corpus Hypercubus)'' is a 1954 oil-on-canvas painting by Salvador Dalí. A nontraditional, surrealism, surrealist Crucifixion in art, portrayal of the Crucifixion, it depicts Christ on a polyhedron net of a tesseract (hypercube). It is one of his best-known paintings from the later period of his career. Background During the 1940s and 1950s Dalí's interest in traditional surrealism diminished and he became fascinated with nuclear science, feeling that "thenceforth, the atom was [his] favorite food for thought". The Atomic bombings of Hiroshima and Nagasaki, atomic bombing at the end of World War II left a lasting impression; his 1951 essay "Mystical Manifesto" introduced an art theory he called "nuclear mysticism" that combined his interests in Catholicism, mathematics, science, and Catalonia, Catalan culture in an effort to reestablish classical values and techniques, which he extensively utilized in ''Corpus Hypercubus''. That same year, to promote nuclear myst ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hexomino
A hexomino (or 6-omino) is a polyomino of order 6; that is, a polygon in the plane made of 6 equal-sized squares connected edge to edge. The name of this type of figure is formed with the prefix hex(a)-. When rotations and reflections are not considered to be distinct shapes, there are 35 different ''free'' hexominoes. When reflections are considered distinct, there are 60 ''one-sided'' hexominoes. When rotations are also considered distinct, there are 216 ''fixed'' hexominoes. Symmetry The figure above shows all 35 possible free hexominoes, coloured according to their symmetry groups: * The twenty grey hexominoes have no symmetry. Their symmetry group consists only of the identity mapping. * The six red hexominoes have an axis of mirror symmetry parallel to the gridlines. Their symmetry group has two elements, the identity and a reflection in a line parallel to the sides of the squares. * The two green hexominoes have an axis of mirror symmetry at 45° to the gridlines. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Two-barred Cross
A two-barred cross is similar to a Latin cross but with an extra bar added. The lengths and placement of the bars (or "arms") vary, and most of the variations are interchangeably called the cross of Lorraine, the patriarchal cross, the Orthodox cross or the archiepiscopal cross. The two bars The two bars can be placed tight together (condensed) or far apart. They can be symmetrically spaced either around the middle, or above or below the middle. One asymmetrical variation has one bar near the top and the other just below the middle. Finally the bars can be of equal length, or with one shorter than the other. Decorations The ends of the arms can be decorated according to different styles. A style with round or rounded ends is called treflée or botonée (from French bouton) in heraldic use. The same style is called budded, apostles' or cathedral cross in religious use. A straight and pointy style called pattée also includes maltese cross variations, and finally a pointed sty ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Salvador Dalí
Salvador Domingo Felipe Jacinto Dalí i Domènech, Marquess of Dalí of Púbol (11 May 190423 January 1989), known as Salvador Dalí ( ; ; ), was a Spanish Surrealism, surrealist artist renowned for his technical skill, precise draftsmanship, and the striking and bizarre images in his work. Born in Figueres in Catalonia, Dalí received his formal education in fine arts in Madrid. Influenced by Impressionism and the Renaissance art, Renaissance masters from a young age, he became increasingly attracted to Cubism and avant-garde movements. He moved closer to Surrealism in the late 1920s and joined the Surrealist group in 1929, soon becoming one of its leading exponents. His best-known work, ''The Persistence of Memory'', was completed in August 1931. Dalí lived in France throughout the Spanish Civil War (1936 to 1939) before leaving for the United States in 1940 where he achieved commercial success. He returned to Spain in 1948 where he announced his return to the Catholic fai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
And He Built A Crooked House
'—And He Built a Crooked House—' is a science fiction short story by American writer Robert A. Heinlein, first published in ''Astounding Science Fiction'' in February 1941. It was reprinted in the anthology '' Fantasia Mathematica'' (Clifton Fadiman, ed.) in 1958, and in the Heinlein collections '' The Unpleasant Profession of Jonathan Hoag'' in 1959 and ''The Best of Robert Heinlein'' in 1973. The story is about a mathematically inclined architect named Quintus Teal who has what he thinks is a brilliant idea to save on real estate costs by building a house shaped like the unfolded net of a tesseract. The title is paraphrased from the nursery rhyme " There Was a Crooked Man". Plot summary Quintus Teal, a "Graduate Architect" in the Los Angeles area, wants architects to be inspired by topology and the Picard–Vessiot theory. During a conversation with friend Homer Bailey he shows models made of toothpicks and clay, representing projections of a four-dimensional tesseract, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nature (journal)
''Nature'' is a British weekly scientific journal founded and based in London, England. As a multidisciplinary publication, ''Nature'' features Peer review, peer-reviewed research from a variety of academic disciplines, mainly in science and technology. It has core editorial offices across the United States, continental Europe, and Asia under the international scientific publishing company Springer Nature. ''Nature'' was one of the world's most cited scientific journals by the Science Edition of the 2022 ''Journal Citation Reports'' (with an ascribed impact factor of 50.5), making it one of the world's most-read and most prestigious academic journals. , it claimed an online readership of about three million unique readers per month. Founded in the autumn of 1869, ''Nature'' was first circulated by Norman Lockyer and Alexander MacMillan (publisher), Alexander MacMillan as a public forum for scientific innovations. The mid-20th century facilitated an editorial expansion for the j ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
