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Maximum number of equidistant points
In mathematics, the equilateral dimension of a metric space is the maximum size of any subset of the space whose points are all at equal distances to
Equilateral_dimension
Invariant measure of fractal dimension
In mathematics, the Hausdorff dimension is a measure of roughness, or more specifically, fractal dimension, that was introduced in 1918 by mathematician
Hausdorff_dimension
Shape containing unit line segments in all directions
Sōichi Kakeya (1917). The minimum area for convex sets is achieved by an equilateral triangle of height 1 and area 1/√3, as Gyula Pál showed in 1920. Kakeya
Kakeya_set
Shape with three equal sides
An equilateral triangle is a triangle with three sides of equal length and three equal angles. It is a regular polygon, occasionally known as the regular
Equilateral_triangle
Geometric space with five dimensions
A five-dimensional (5D) space is a mathematical or physical space that has five independent dimensions. In physics and geometry, such a space extends
Five-dimensional_space
Four-dimensional analogue of the cube
than a zero-dimensional point) that is radially equilateral. The longest vertex-to-vertex diagonal of an n {\displaystyle n} -dimensional hypercube of
Tesseract
Topics referred to by the same term
Equilateral dimension of a metric space, in mathematics Equilateral triathlon, in which each leg would take an approximately equal time Venus Equilateral, a set
Equilateral_(disambiguation)
Real-valued number of spatial dimensions
fractal dimension that build on this basic concept of change in detail with change in scale, see § Examples below. Ultimately, the term fractal dimension became
Fractal_dimension
Pentagon with all sides equal but the angles may not be equal
In geometry, an equilateral pentagon is a polygon in the Euclidean plane with five sides of equal length. Its five vertex angles can take a range of sets
Equilateral_pentagon
Geometric concept
small amounts to try to minimise the polynomial in terms of the y. Equilateral dimension Spherical code Soddy's hexlet Cylinder sphere packing Conway, John
Kissing_number
Shape with six sides
hexagon is 720°. A regular hexagon is defined as a hexagon that is both equilateral and equiangular. Its internal angle is one-third of a circle, equal to
Hexagon
Infinitely detailed mathematical structure
arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales
Fractal
Quadrilateral with sides of equal length
In geometry, a rhombus (pl.: rhombi or rhombuses) is an equilateral quadrilateral, a quadrilateral whose four sides all have the same length. Other names
Rhombus
Set of distances defined from a set of points
element. Kusner's conjecture states that the equilateral dimension of a d {\displaystyle d} -dimensional space with the Manhattan distance is exactly
Distance_set
Solid with four equal triangular faces
four equilateral triangular faces. A regular tetrahedron is a tetrahedron (that is, a four-sided polyhedron) in which all four faces are equilateral triangles
Regular_tetrahedron
Topics referred to by the same term
the set as a function of the box size Equilateral dimension of a metric space (also called the metric dimension), the maximum number of points at equal
Metric_dimension
Polyhedron with eight triangular faces
special case is the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Many types of irregular
Octahedron
Fractal composed of triangles
a fractal with the overall shape of an equilateral triangle, subdivided recursively into smaller equilateral triangles. Originally constructed as a Sierpiński
Sierpiński_triangle
Geometric object with flat sides
generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions n as an n-dimensional polytope or
Polytope
Theory of subatomic structure
which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings move
String_theory
Size of a two-dimensional surface
with a given perimeter is equilateral. The triangle of largest area of all those inscribed in a given circle is equilateral; and the triangle of smallest
Area
Pyramid with a square base
all of the pyramid's edges are equal in length, its triangles are all equilateral, an example of a Johnson solid. Square pyramids have appeared throughout
Square_pyramid
Hausdorff-Besicovitch dimension strictly exceeds the topological dimension." Presented here is a list of fractals, ordered by increasing Hausdorff dimension, to illustrate
List of fractals by Hausdorff dimension
List_of_fractals_by_Hausdorff_dimension
Polyhedron with 8 triangles and 6 squares
four-dimensional 24-cell and 8-cell (tesseract). Radially equilateral polytopes are those that can be constructed, with their long radii, from equilateral
Cuboctahedron
1884 novella by Edwin Abbott Abbott
Written pseudonymously by "A Square", the book used the fictional two-dimensional world of Flatland to satirise the class and gender hierarchies of Victorian
Flatland
Shape with three sides
length is an equilateral triangle. And a triangle with all sides having different lengths is a scalene triangle. Isosceles triangle Equilateral triangle Scalene
Triangle
Curved triangle with constant width
constructed either directly from three circles, or by rounding the sides of an equilateral triangle. The three-circle construction may be performed with a compass
Reuleaux_triangle
Geometric problems involving the partition of a figure
posed by puzzle writer Henry Dudeney in 1902. It seeks a dissection from equilateral triangle into a square. Dudeney provided a hinged dissection with four
Dissection_problem
Fractal curve
an equilateral triangle, and each successive stage is formed by adding outward bends to each side of the previous stage, making smaller equilateral triangles
Koch_snowflake
Polyhedron formed by joining two prisms
These prisms cover the square faces so the resulting polyhedron has four equilateral triangles and four squares, making eight faces in total, an octahedron
Gyrobifastigium
Natural number
the two-dimensional orthographic projection of the four-dimensional 8-8 duoprism. An octahedron is a regular polyhedron with eight equilateral triangles
8
Constructible polygon Convex polygon Cyclic polygon Equiangular polygon Equilateral polygon Penrose tile Polyform Regular polygon Simple polygon Tangential
List of two-dimensional geometric shapes
List_of_two-dimensional_geometric_shapes
Non-orientable surface with one edge
symmetrically by five of the ten equilateral triangles of a four-dimensional regular simplex. This four-dimensional polyhedral Möbius strip is the only
Möbius_strip
Form of an object
Each of these is divided into smaller categories; triangles can be equilateral, isosceles, obtuse, acute, scalene, etc. while quadrilaterals can be
Shape
Shape that is the intersection of two circles with the same radius
Euclid's Elements, where it forms the first step in constructing an equilateral triangle using a compass and straightedge. The triangle has as its vertices
Vesica_piscis
Path that surrounds an area
2w+2\ell .} An equilateral polygon is a polygon which has all sides of the same length (for example, a rhombus is a 4-sided equilateral polygon). To calculate
Perimeter
Kepler–Poinsot polyhedron with 20 faces
pentagram, its two-dimensional analogue, via the extension of the (n–1)-dimensional simplex faces of the core n-polytope (equilateral triangles for the
Great_icosahedron
particular solution to the model instance. There are two types of two-dimensional cell shapes that are commonly used. These are the triangle and the quadrilateral
Types_of_mesh
Integer associated with a graph
Euclidean space of dimension n − 1 {\displaystyle n-1} . For example, it takes two dimensions to immerse K 3 {\displaystyle K_{3}} (an equilateral triangle),
Dimension_(graph_theory)
Nine-pointed star polygon
from the regular enneagon points but connected as a compound of three equilateral triangles. (If the triangles are alternately interlaced, this results
Enneagram_(geometry)
Multi-dimensional generalization of triangle
polytope in any given dimension. For example, a 0-dimensional simplex is a point, a 1-dimensional simplex is a line segment, a 2-dimensional simplex is a triangle
Simplex
Schläfli symbol {}. Polygon Equilateral Cyclic polygon Convex polygon Star polygon Pentagram Regular polygon Equilateral triangle Simplex Square Cross-polytope
List_of_mathematical_shapes
Plane fractal built from squares
extended to other shapes. For instance, subdividing an equilateral triangle into four equilateral triangles, removing the middle triangle, and recursing
Sierpiński_carpet
Two tetrahedra crossing each other
extension of the hexagram: the hexagram is a two-dimensional shape formed from two crossing equilateral triangles, centrally symmetric to each other, and
Stellated_octahedron
Two-dimensional packing problem
possible equilateral triangle which an amount n of unit circles can be packed into? More unsolved problems in mathematics Circle packing in an equilateral triangle
Circle packing in an equilateral triangle
Circle_packing_in_an_equilateral_triangle
Six-pointed star polygon
2{3}, or {{3}}. The term is used to refer to a compound figure of two equilateral triangles. The intersection is a regular hexagon. The hexagram is part
Hexagram
Group of transformations under which the object is invariant
is isomorphic to the Klein four-group, is the symmetry group of a non-equilateral rectangle. This figure has four symmetry operations: the identity operation
Symmetry_group
Plane figure bounded by line segments
Equiangular: all corner angles are equal. Equilateral: all edges are of the same length. Regular: both equilateral and equiangular. Cyclic: all corners lie
Polygon
Triangle in hyperbolic geometry
arbitrary dimension always lie on the same plane. Hence planar hyperbolic triangles also describe triangles possible in any higher dimension of hyperbolic
Hyperbolic_triangle
Periodic set of points
More abstractly, a lattice can be described as a free abelian group of dimension n {\displaystyle n} which spans the vector space R n {\displaystyle
Lattice_(group)
Two tetrahedra joined by one face
these tetrahedra are regular, all faces of a triangular bipyramid are equilateral. It is an example of a deltahedron, composite polyhedron, and Johnson
Triangular_bipyramid
Additive process used to make a 3D object
also called additive manufacturing, is the construction of a three-dimensional object from a CAD model or a digital 3D model. It can be done in a variety
3D_printing
Quadrilateral with two pairs of parallel sides
Parallelogram --sides, angles and slope Area of Parallelogram at cut-the-knot Equilateral Triangles On Sides of a Parallelogram at cut-the-knot Definition and
Parallelogram
Four-dimensional analogue of the tetrahedron
tetrahedron. This cannot be done in 3-dimensional space. The regular 5-cell is a solution to the problem: Make 10 equilateral triangles, all of the same size
5-cell
Archimedean solid with 8 faces
tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 equilateral triangle faces, 12 vertices and 18 edges (of two types). It can be constructed
Truncated_tetrahedron
Mathematical treatise by Euclid
explicitly as axioms. For example, in the first construction of Book 1, of an equilateral triangle, Euclid used a premise that was neither postulated nor proved:
Euclid's_Elements
Cylinder whose generatrices are perpendicular to the bases
be visualized using online graphing calculators such as Desmos. The equilateral cylinder is characterized by being a right circular cylinder in which
Right_circular_cylinder
Convex polyhedron with 16 triangular faces
with four equilateral triangles, so that the resulting polyhedron has 16 equilateral triangles as its faces. A polyhedron with only equilateral triangles
Gyroelongated square bipyramid
Gyroelongated_square_bipyramid
Polyform whose base form is an equilateral triangle
sometimes triangular polyomino) is a polyform whose base form is an equilateral triangle. The word polyiamond is a back-formation from diamond, because
Polyiamond
Puzzle involving the arrangement of matchsticks
thinking is to form four equilateral triangles from six matches; this can only be done by arranging the matches in a three-dimensional pyramid shape. Publishing
Matchstick_puzzle
Solid with eight equal triangular faces
In geometry, a regular octahedron is an eight-sided polyhedron with equilateral triangles as its faces. Known for its highly symmetrical form, the regular
Regular_octahedron
Construction for n-dimensional noise functions
dimensions has only n + 1 {\displaystyle n+1} corners. The triangles are equilateral in 2D, but in higher dimensions the simplices are only approximately
Simplex_noise
Problem in geometric probability
be represented as the three barycentric coordinates of a point in an equilateral triangle whose height is the length of the original line segment, or
Broken_stick_problem
10th Johnson solid (13 faces)
pyramid is the Johnson solid that can be constructed by attaching an equilateral square pyramid to a square antiprism. It occurs in chemistry; for example
Gyroelongated_square_pyramid
Problems which attempt to find the most efficient way to pack objects into containers
triangle - good estimates are known for n < 300. Packing circles in an equilateral triangle - Optimal solutions are known for n ≤ 15, and conjectures are
Packing_problems
Algorithm used for points in euclidean space
well shaped; in the case of triangles, often elements that are nearly equilateral triangles are preferred. Lloyd's algorithm can be used to smooth a mesh
Lloyd's_algorithm
Solid with twenty equal triangular faces
or by putting points onto the cube. The resulting polyhedron has 20 equilateral triangles as its faces, 30 edges, and 12 vertices. It is an example of
Regular_icosahedron
Convex polyhedron with 14 triangle faces
convex polyhedron with 14 equilateral triangles as its faces. It can be constructed from a triangular prism by attaching equilateral square pyramids to each
Triaugmented_triangular_prism
Multi-stage paper folding technique
any equilateral polyhedron including those having rhombic faces, like the rhombic dodecahedron. The Mukhopadhyay module can form any equilateral polyhedron
Modular_origami
Limiting case which is different from the rest of the class
are degenerate. For example, right triangles, isosceles triangles and equilateral triangles are non-generic and non-degenerate. In fact, degenerate cases
Degeneracy_(mathematics)
25th Johnson solid (37 faces)
pentagonal rotunda is a Johnson solid, a convex polyhedron with thirty equilateral triangles, six regular pentagons, and one regular decagon as its faces
Gyroelongated pentagonal rotunda
Gyroelongated_pentagonal_rotunda
Polyhedron with four faces
A regular tetrahedron is a tetrahedron in which all four faces are equilateral triangles. In other words, all of its faces are the same size and shape
Tetrahedron
Natural number, composite number
28,32,38\}} where 14 is the seventh such number. There are fourteen equilateral triangles formed by the sides and diagonals of a regular six-sided hexagon
14_(number)
Barycentric plot on three variables
graphically depicts the ratios of the three variables as positions in an equilateral triangle. It is used in physical chemistry, petrology, mineralogy, metallurgy
Ternary_plot
Figurate number
integers that can be represented as a lattice of points arranged in an equilateral triangle. The triangular lattice representing the n {\displaystyle n}
Triangular_number
Two-dimensional fractal
based on recursively drawing equilateral triangles and the Sierpinski carpet." The T-square fractal has a fractal dimension of ln(4)/ln(2) = 2.[citation
T-square_(fractal)
Topics referred to by the same term
perspective"), a method for drawing three-dimensional objects on flat paper so that a cubical grid is projected onto an equilateral triangle grid and distances aligned
Isometric
Three linked but pairwise separated rings
artworks with three equilateral triangles made out of sheet metal, linked to form Borromean rings and resembling a three-dimensional version of the valknut
Borromean_rings
Symbol of ten points laid in four rows
perfect eleventh), is not. The Tetractys [also known as the decad] is an equilateral triangle formed from the sequence of the first ten numbers aligned in
Tetractys
Archimedean solid with 14 faces
Archimedean solid that arises from a regular octahedron by removing six equilateral square pyramids, one at each of the octahedron's vertices. The truncated
Truncated_octahedron
Point in a triangle that can be seen as its middle under some criteria
under reflection and so fail to qualify as triangle centers. For an equilateral triangle, all triangle centers coincide at its centroid. However, the
Triangle_center
Triangular prism attached by a square pyramid
augmented triangular prism is a polyhedron constructed by attaching an equilateral square pyramid onto one of the square faces of a triangular prism. As
Augmented_triangular_prism
Line constructed from a triangle
(/ˈɔɪlər/ OY-lər), is a line determined from any triangle that is not equilateral. It is a central line of the triangle, and it passes through several
Euler_line
Right-angled non-convex polyhedron
{\displaystyle 4} . The faces of the icosahedron are eight congruent equilateral triangles with the short side length, and twelve congruent obtuse isosceles
Jessen's_icosahedron
Prism with a 3-sided base
all edges are equal in length, its bases and its lateral faces are all equilaterals and squares, respectively. Hence, the triangular prism is semiregular
Triangular_prism
Natural number
primes (5, 13, 563). A shape with five sides is called a pentagon. The equilateral pentagon is the first regular polygon that does not tile the plane with
5
Complex structures in matter physics
is shown in Figure 1. Three magnetic ions reside on the corners of an equilateral triangle with antiferromagnetic interactions between them; the energy
Geometrical_frustration
Plane curve: conic section
case a = b {\displaystyle a=b} the hyperbola is called rectangular (or equilateral), because its asymptotes intersect at right angles. For this case, the
Hyperbola
regular faces. The first six Johnson solids satisfy this criterion: the equilateral square pyramid, pentagonal pyramid, triangular cupola, square cupola
List_of_Johnson_solids
Mathematical model of the physical space
∘ {\displaystyle \alpha +\beta +\gamma =180^{\circ }} This means an equilateral triangle has three 60° interior angles, and every triangle has at least
Euclidean_geometry
Geometry with 7 points and 7 lines
its seven points as the vertices, edge midpoints, and centroid of an equilateral triangle, and its seven lines as the three sides and three symmetry axes
Fano_plane
Classification of a two-dimensional repetitive pattern
plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns
Wallpaper_group
Ancient geometric triangle
The Dragon's Eye is an isosceles or equilateral triangle pointing downward, with a "Y" in the middle connecting the three points of the triangle together
Dragon's_Eye_(symbol)
polytopes of rank 2 (2-polytopes) are called polygons. Regular polygons are equilateral and cyclic. A p-gonal regular polygon is represented by Schläfli symbol
List_of_regular_polytopes
Historical building in Singapore
applied to civic and commercial building by the colonial government. The equilateral triangle at the top of the façade with decorations on the edges and the
Queen's_Theatre,_Singapore
Mathematical problem
of two unit equilateral triangles joined at a common vertex, x. Each of these triangles is joined along another edge to another equilateral triangle; the
Hadwiger–Nelson_problem
Movement of an object which leaves at least one point unchanged
2001 [1994] Product of Rotations at cut-the-knot. When a Triangle is Equilateral at cut-the-knot. Rotate Points Using Polar Coordinates, howtoproperly
Rotation
Doughnut-shaped surface of revolution
is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanar with the circle
Torus
Polyhedron with parallel bases connected by triangles
k ≤ 2n – 1; if the n-antiprism is uniform (i.e. if the triangles are equilateral), then: 2 h 2 = cos π n − cos 2 π n . {\displaystyle 2h^{2}=\cos
Antiprism
Regular star 4-polytope with 600 faces
two-dimensional analogue, via the extension of said (n-1)-D simplex faces of the core nD polytope (tetrahedra for the grand 600-cell, equilateral triangles
Grand_600-cell
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