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Tiling by squares of two sizes
A Pythagorean tiling or two squares tessellation is a tiling of the Euclidean plane by squares of two different sizes, in which each square touches four
Pythagorean_tiling
Covering by shapes without overlaps or gaps
wallpaper groups. A tiling that lacks a repeating pattern is called "non-periodic". An aperiodic tiling uses a small set of tile shapes that cannot form
Tessellation
Relation between sides of a right triangle
culture Pythagorean expectation Pythagorean hodograph curve – determined by polynomials that obey the Pythagorean theorem Pythagorean tiling Rational
Pythagorean_theorem
Semiregular tiling
In geometry, the truncated square tiling is a semiregular tiling by regular polygons of the Euclidean plane with one square and two octagons on each vertex
Truncated_square_tiling
The chiral forms be seen as two overlapping pythagorean tilings. The dual tiling looks like a square tiling with half of the squares divided into central
Chamfered_square_tiling
Shape with four equal sides and angles
of a square tiling form a square lattice. Squares of more than one size can also tile the plane, for instance in the Pythagorean tiling, named for its
Square
Mathematics book
topics in tiling theory: colored patterns and tilings, polygonal tilings, aperiodic tilings, Wang tiles, and tilings with unusual kinds of tiles. Each chapter
Tilings_and_patterns
Geometry problem on tiling by hypercubes
Keller graphs. The related Minkowski lattice cube-tiling conjecture states that whenever a tiling of space by identical cubes has the additional property
Keller's_conjecture
Generalisation of dice with identical faces
tiling (m = 1) has congruent faces, either directly or reflectively, which occur in one or more symmetry positions. An m-hedral polyhedron or tiling has
Isohedral_figure
Vertex-transitive tiling of the plane by regular polygons
tiles meet edge-to-edge can be relaxed, allowing additional tilings such as the Pythagorean tiling. Symmetry group triangles with retrogrades include: (4/3
Uniform_tiling
Classification of a two-dimensional repetitive pattern
reflections Viennese cane Renaissance earthenware Pythagorean tiling Generated from a photograph 4 co-uniform tiling Orbifold signature: *442 Coxeter notation:
Wallpaper_group
List of appearances of Pythagoras in art and pop culture
endeavors, but also as an example of abstruse higher learning in general. Pythagorean tiling has been used as proofs by the 9th-century Islamic mathematicians
Pythagoras_in_popular_culture
Topics referred to by the same term
Hopscotch hashing, in computer programming Hopscotch pattern or Pythagorean tiling, a floor tile layout This disambiguation page lists articles associated with
Hopscotch_(disambiguation)
Natural number
Kevin. "Shield - a 3.7.42 tiling". Imperfect Congruence. Retrieved 2023-01-09. 3.7.42 as a unit facet in an irregular tiling. Poonen, Bjorn; Rubinstein
7
British astronomer and mathematician (1801–1898)
generated by overlaying a regular square tiling whose prototile is the larger square with a Pythagorean tiling generated by the two smaller squares. Perigal
Henry_Perigal
Philosophical argument
square tiling of the plane representing a discrete space. A discretized triangle, n units tall and n units long, can be constructed on the tiling. The hypotenuse
Weyl's_tile_argument
Polyhedron with regular congruent polygons as faces
paracompact regular honeycombs have Euclidean tiling facets and vertex figures that act like finite polyhedra. Such tilings have an angle defect that can be closed
Regular_polyhedron
Solid with four equal triangular faces
from any vertex to the midpoint of an edge, and by the calculation of Pythagorean theorem, the height of any equilateral triangle is 3 2 a {\textstyle
Regular_tetrahedron
Natural number
tiling. This tiling is one of eight Archimedean tilings that are semi-regular, or made of more than one type of regular polygon, and the only tiling that
8
Painting by Jacob Ochtervelt
the interior, the Pythagorean tiling of its flooring, has been called out as an example of the long history of use of this tiling pattern. Street Musicians
Street_Musicians_at_the_Door
Any of the five regular polyhedra
which exactly cover the sphere. Spherical tilings provide two infinite additional sets of regular tilings, the hosohedra, {2,n} with 2 vertices at the
Platonic_solid
Persian mathematician and astronomer
astronomer al-Marwazi before him. He gave a proof of the Pythagorean theorem using the Pythagorean tiling. Al-Nayrizi gave a mathematical proof of the parallel
Al-Nayrizi
Plane fractal constructed from squares
right triangle, in a configuration traditionally used to depict the Pythagorean theorem. If the largest square has a size of L × L, the entire Pythagoras
Pythagoras_tree_(fractal)
Overview of and topical guide to geometry
Sangaku Straightedge Symmedian Tessellation Prototile Aperiodic tiling Wang tile Penrose tiling Trapezoid (trapezium) Isosceles trapezoid Triangle Acute and
Outline_of_geometry
Five-pointed star polygon
measures of the human body', and an 'inverted' (point-down) version of the Pythagorean 'hygeia' pentagram in the section on 'characters, received only by revelation
Pentagram
Swedish mathematician
the board game Go. One of the many proofs of the Pythagorean theorem based on the Pythagorean tiling is sometimes called "Olof Hanner's Jigsaw Puzzle"
Olof_Hanner
Polar circle (geometry) Pompeiu's theorem Pons asinorum Pythagorean theorem Inverse Pythagorean theorem Reuleaux triangle Regiomontanus Regiomontanus'
List_of_triangle_topics
Shape with three equal sides
tessellation's instances are the triangular tiling where six equilateral triangles surrounds a common vertex, and the sphinx tiling as a special case of the polyiamond
Equilateral_triangle
Type of non-Euclidean geometry
"hyperbolic soccerball" (more precisely, a truncated order-7 triangular tiling). Instructions on how to make a hyperbolic quilt, designed by Helaman Ferguson
Hyperbolic_geometry
Natural number
the smallest integer-sided right triangle, making part of the smallest Pythagorean triple (3, 4, 5). 5 is the first safe prime and the first good prime
5
2002 Japanese TV program
and between each corner (segment), there are Pythagorean Devices (ピタゴラ装置, Pitagora Sōchi). "Pythagorean device" is the equivalent Japanese colloquialism
PythagoraSwitch
Natural number
134. "Shield - a 3.7.42 tiling". Kevin Jardine's projects. Kevin Jardine. Retrieved 7 March 2022. "Dancer - a 3.8.24 tiling". Kevin Jardine's projects
17_(number)
Number, approximately 1.618
can be used as the prototiles for a form of the Penrose tiling. The rhombic Penrose tiling contains two types of rhombus, a thin rhombus with angles
Golden_ratio
1619 book by Johannes Kepler
establish his celestial-harmonic relationships was the abandonment of the Pythagorean tuning as the basis for musical consonance and the adoption of geometrically
Harmonice_Mundi
Tiling puzzle where pieces can be assembled in different ways
dissection puzzle, also called a transformation puzzle or Richter puzzle, is a tiling puzzle where a set of pieces can be assembled in different ways to produce
Dissection_puzzle
Branch of mathematics
theorem. Pythagoras established the Pythagorean School, which is credited with the first proof of the Pythagorean theorem, though the statement of the
Geometry
Generalization of golden and silver ratios
\theta } is a positive integer, as it is with some Pythagorean triangles. For a primitive Pythagorean triple, a2 + b2 = c2, with positive integers a < b
Metallic_mean
Isogonal polyhedron with regular faces
Semiregular polyhedron Polyhedron model Pseudo-uniform polyhedron Uniform tiling Uniform tilings in hyperbolic plane Diudea (2018), p. 40. Coxeter, Longuet-Higgins
Uniform_polyhedron
Solid with 12 equal pentagonal faces
further convex regular polyhedra. Iamblichus states that Hippasus, a Pythagorean, perished in the sea, because he boasted that he first divulged "the
Regular_dodecahedron
2011 book by Roger B. Nelsen and Claudi Alsina
triangles, star polygons, Platonic solids, and figurate numbers The Pythagorean theorem, Thales's theorem on right triangles in semicircles, and geometric
Icons_of_Mathematics
Number used to approximate the square root of 2
lengths a, b, c (necessarily satisfying the Pythagorean theorem a2 + b2 = c2), then (a,b,c) is known as a Pythagorean triple. As Martin (1875) describes, the
Pell_number
Historical development of geometry
expression of the Pythagorean Theorem in the world, although it had already been known to the Old Babylonians." They make use of Pythagorean triples, which
History_of_geometry
Golden Triangle Obtuse triangle Rational triangle Heronian triangle Pythagorean triangle Isosceles heronian triangle Primitive Heronian triangle Right
List of two-dimensional geometric shapes
List_of_two-dimensional_geometric_shapes
Solid with six equal square faces
{\displaystyle a{\sqrt {3}}} . Both formulas can be determined by using the Pythagorean theorem. The surface area of a cube A {\displaystyle A} is six times
Cube
Type of geometry
A hyperbolic triheptagonal tiling in a Beltrami–Klein model projection
Projective_geometry
Natural number
hypotenuse of the primitive Pythagorean triple ( 11 , 60 , 61 ) {\displaystyle (11,60,61)} . There are sixty-one 3-uniform tilings. Sixty-one is the exponent
61_(number)
Dissection puzzle
is no reason to suspect that tangrams were used in the proof of the Pythagorean theorem, as is sometimes reported, it is likely that this style of geometric
Tangram
Ideas from Mathematics have been used as inspiration for fiber arts
the Koch curve, the Clifford torus, San Gaku, Mascheroni's cardioid, Pythagorean triples, spidrons, and the six trigonometric functions. Knitted mathematical
Mathematics_and_fiber_arts
Number, approximately 2.41421
The silver ratio appears prominently in the Ammann–Beenker tiling, a non-periodic tiling of the plane with octagonal symmetry, build from a square and
Silver_ratio
1890 BC). All these texts mention the so-called Pythagorean triples, so, by inference, the Pythagorean theorem seems to be the most ancient and widespread
History_of_mathematics
Branch of geometry that studies combinatorial properties and constructive methods
A tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics
Discrete_geometry
Non-Euclidean geometry
elliptic geometry is also self-consistent and complete. Elliptic tiling Spherical tiling Duncan Sommerville (1914) The Elements of Non-Euclidean Geometry
Elliptic_geometry
short tons). An aperiodic tiling was considered, to avoid the rhythm of a structural grid, but in practice a Penrose tiling was too complex, so a grid
Mathematics_and_architecture
Catalan solid with 24 faces
central point. Disdyakis triacontahedron Disdyakis dodecahedron Kisrhombille tiling Compound of three octahedra Deltoidal icositetrahedron, another 24-face
Tetrakis_hexahedron
Roman-era word square with a Latin palindrome
Christians. Other less-supported academic origin theories include a Pythagorean or Stoic puzzle, a Gnostic or Orphic or Italian pagan amulet, a cryptic
Sator_Square
Comune in Calabria, Italy
Cylon, during which many Pythagoreans were massacred and Pythagoras himself had to flee to Metapontum, led to the Pythagoreans being driven out and a democracy
Crotone
Triangle with at least two sides congruent
{a^{2}-{\frac {b^{2}}{4}}}}.} This formula can also be derived from the Pythagorean theorem using the fact that the altitude bisects the base and partitions
Isosceles_triangle
Infinitely detailed mathematical structure
rep-tiled into pieces each scaled down by a scale-factor of 1/r, there are a total of rn pieces. Now, consider the Koch curve. It can be rep-tiled into
Fractal
Number that represents a hexagon with a dot in the center
rocket tubes. Hindin, H. J. (1983). "Stars, hexes, triangular numbers and Pythagorean triples". J. Rec. Math. 16: 191–193. Deza, Elena; Deza, M. (2012). Figurate
Centered_hexagonal_number
as a parallelohedron? Does every higher-dimensional tiling by translations of convex polytope tiles have an affine transformation taking it to a Voronoi
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Natural number
41. It is a Jacobsthal-Lucas number and the hypotenuse of a primitive Pythagorean triangle. It is a Proth number because 1025 = 210 + 1. It is a member
1000_(number)
Temple on the Athenian Acropolis, Greece
endeavoured to incorporate the idea that the Parthenon’s design reflects Pythagorean musical ratios, such as 3:2 (the perfect fifth) and 4:9. According to
Parthenon
2008 science fiction novel by Neal Stephenson
stories from Penrose's The Road to Reality; and the theory of aperiodic tilings, which appear in the Teglon puzzle in the novel. Stephenson also cites
Anathem
Chess problems Leaper that makes moves 5 units in length. Due to the Pythagorean theorem, it has twelve possible directions. Also named Root-25-Leaper
List_of_fairy_chess_pieces
Screen resolution measured in pixels per length
done in two steps: Calculate diagonal resolution in pixels using the Pythagorean theorem: d p = w p 2 + h p 2 {\displaystyle d_{p}={\sqrt {w_{p}^{2}+h_{p}^{2}}}}
Pixel_density
ones), one can prove that the Pythagorean theorem holds (that is, one can generate the string corresponding to the Pythagorean theorem). According to formalism
Philosophy_of_mathematics
Tabletop game
of the Pythagorean Theorem that's been color-coded to reflect bearing angles used by the AVID, and tilt blocks, box miniatures and stacking tiles for on-map
Attack_Vector:_Tactical
Desargues's projective geometry. A persistent view, based ultimately on the Pythagorean notion of harmony in music, holds that everything was arranged by Number
Mathematics_and_art
Special case of the logarithmic spiral Spiral of Theodorus (also known as Pythagorean spiral) c. 500 BC Contiguous right triangles composed of one leg with
List_of_spirals
Imperial dynasty in China (202 BC – 220 AD)
accurate approximations for pi, providing mathematical proof of the Pythagorean theorem, use of the decimal fraction, Gaussian elimination to solve linear
Han_dynasty
Algorithm for computing greatest common divisors
terms of the tiling analogy given above for the greatest common divisor. Assume that we wish to cover an a×b rectangle with square tiles exactly, where
Euclidean_algorithm
Person with an extensive knowledge of mathematics
mathematicians grew when Pythagoras of Samos (c. 582 – c. 507 BC) established the Pythagorean school, whose doctrine it was that mathematics ruled the universe and
Mathematician
Ancient Amorite-Akkadian state in Mesopotamia
to seven places (YBC 7289). They also demonstrated knowledge of the Pythagorean theorem well before Pythagoras. The ner of 600 and the sar of 3600 were
Babylonia
Ancient Greek god of the sea, earthquakes, and horses
Troezen in the Peloponnese, part of the modern island-pair Poros. Early roof tiles from c.650 BC suggest the existence of a precursor to the Late Archaic temple
Poseidon
Numbers obtained by adding the two previous ones
triangle with integer sides, or in other words, the largest number in a Pythagorean triple, obtained from the formula ( F n F n + 3 ) 2 + ( 2 F n + 1 F n
Fibonacci_sequence
Traditionally Chinese religious and philosophical tradition
(with the phoenix also standing for yin) made from multicolored ceramic tiles. In general though, Chinese Taoist architecture lacks universal features
Taoism
Property of a mathematical space
Polygon Net Complex number Cartesian coordinate system List of uniform tilings Area 3 dimensions Platonic solid Polyhedron Stereoscopy (3-D imaging) 3-manifold
Dimension
Positive real number which when multiplied by itself gives 5
rectangle whose sides are of length 1 and 2, as is evident from the Pythagorean theorem. Such a rectangle can be obtained by halving a square, or by
Square_root_of_5
4th-century Alexandrian astronomer and mathematician
pronunciation [y.pa.ˈti.a] Using music to relieve lustful urges was a Pythagorean remedy stemming from an anecdote from the life of Pythagoras relating
Hypatia
Cradle of civilization in North Africa
two-thirds is shown on the right. Ancient Egyptian mathematicians knew the Pythagorean theorem as an empirical formula. They were aware, for example, that a
Ancient_Egypt
theory of light traveling into the eye (and not vice versa like in Pythagoreanism), since the Mojing states that the reflected light shining forth from
List_of_Chinese_inventions
mathematician – Pythagorean theorem, Pythagorean triple, Pythagorean tuning, Pythagorean expectation, Pythagorean hammers, Pythagorean trigonometric identity
List_of_eponyms_(L–Z)
Mathematical invariance under transformations
floor plans, and down to the design of individual building elements such as tile mosaics. Islamic buildings such as the Taj Mahal and the Lotfollah mosque
Symmetry
Zaragoza 1 1994 Les Pythagoriciens (The Pythagoreans) 1994 Praça de Moscovo (Moscow Square) 1995 Paris reproduced on a tile panel in 1998; not related to the
Nadir_Afonso_artworks
Polyhedron with four faces
Solutions, Crux Mathematicorum, 11 (5): 162–166, May 1985 Wacław Sierpiński, Pythagorean Triangles, Dover Publications, 2003 (orig. ed. 1962), p. 107. Note however
Tetrahedron
rex 1975 Jul On tessellating the plane with convex polygon tiles 1975 Aug More about tiling the plane: the possibilities of polyominoes, polyiamonds, and
List of Martin Gardner Mathematical Games columns
List_of_Martin_Gardner_Mathematical_Games_columns
Branch of mathematics
and dynamical information. This approach is important for foliations, tilings, dynamical systems and examples from mathematical physics. A smooth compact
Noncommutative_geometry
Four-dimensional analog of the octahedron
link. The 16 triangle faces can be seen in a 2D net within a triangular tiling, with 6 triangles around every vertex. The purple edges represent the Petrie
16-cell
Ancient city on the Ionian Sea
philosophers Echecrates, Timaeus, and Acrion, founders of a flourishing Pythagorean school (introduced to Locri at the time of Dionysius I). Plato visited
Epizephyrian_Locris
Goddess from Greek mythology, wife and sister of Zeus
temples, a forest of 155 columns is visible. There is also no evidence of tiles on this temple, suggesting either the temple was never finished or that
Hera
Roman architectural features
The dome may use rows of 28 coffers because 28 was considered by the Pythagoreans to be a perfect number. Hadrian was an amateur architect and it was apparently
2nd-century_Roman_domes
Sources of ancient myth
isles. Olaus Magnus on his Carta Marina identifies the location, calling it Tile. See, image of Bellenden's edition in main entry. There are a number of editions
Barnacle_goose_myth
Mathematical board game
Games, pp. 332–342, ISBN 0-19-212998-8 J.F.C. Richards, Boissiere’s Pythagorean game, Scripta Mathematica 12(1946)177-217. David Sepkoski, "Ann E. Moyer:
Rithmomachia
historical and notable women whose names are displayed on the handmade white tiles of the Heritage Floor as part of Judy Chicago's The Dinner Party art installation
List of women in the Heritage Floor
List_of_women_in_the_Heritage_Floor
Aspect of women's history
hand-to-hand combat, and defeated the Illyrian army. 4th century BCE – Pythagorean philosopher, Timycha, was captured by Sicilian soldiers during a battle
Women_in_ancient_warfare
Property of objects which are scaled or mirrored versions of each other
the geometric mean theorem, Ceva's theorem, Menelaus's theorem and the Pythagorean theorem. Similar triangles also provide the foundations for right triangle
Similarity_(geometry)
Visible regularity of form found in the natural world
Browne discussed "how Nature Geometrizeth" in The Garden of Cyrus, citing Pythagorean numerology involving the number 5, and the Platonic form of the quincunx
Patterns_in_nature
nanotechnology Theano (6th century BC), one or possibly two different Pythagorean philosophers Diana Thomas, American mathematician who studies nutrition
List_of_women_in_mathematics
depict Confucian ideals, such as the famous "Painted Basket" and a roof tile imprinted with a Confucian institution's symbols, have been found in the
Korean_Confucianism
Special mathematical function
{\displaystyle iK'} etc.) that uniquely determine a tiling of the plane by parallelograms. The tiling is referred to as the modular symmetry given by the
Nome_(mathematics)
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PYTHAGOREAN TILING
PYTHAGOREAN TILING
PYTHAGOREAN TILING
PYTHAGOREAN TILING
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PYTHAGOREAN TILING
PYTHAGOREAN TILING
PYTHAGOREAN TILING
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