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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
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
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
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)
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Size of a two-dimensional surface
It is the two-dimensional analogue of the length of a curve (a one-dimensional concept) or the volume of a solid (a three-dimensional concept). Two different
Area
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
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
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
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
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
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
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)
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
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
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)
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 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 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
Path that surrounds an area
boundary that encompasses, surrounds, or outlines either a two-dimensional shape or a one-dimensional line. The perimeter of a circle or an ellipse is called
Perimeter
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
Schläfli symbol {}. Polygon Equilateral Cyclic polygon Convex polygon Star polygon Pentagram Regular polygon Equilateral triangle Simplex Square Cross-polytope
List_of_mathematical_shapes
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
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
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
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
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
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
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
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
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
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. It is also
Truncated_octahedron
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
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
Five-pointed star polygon
(freq. applied to a hexagram formed by two intersecting or interlaced equilateral triangles)." πένταλφα, "five Alphas", interpreting the shape as five
Pentagram
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
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
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
Conic solid with a polygonal base
triangles. If the edges are all equal in length, such that its faces are equilateral, and one of them is considered as the base, it is known as a regular
Pyramid_(geometry)
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
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)
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
Problems which attempt to find the most efficient way to pack objects into containers
different objects with their sizes specified, or a single object of a fixed dimension that can be used repeatedly. Usually the packing must be without overlaps
Packing_problems
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)
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
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)
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
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
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
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
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
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)
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
Modular origami module
interconnected Sonobe units will form an open-bottomed triangular pyramid with an equilateral triangle for the open bottom, and isosceles right triangles as the other
Sonobe
Type of dynamical system in mathematics
a regular polygon has almost every orbit periodic. The cases of the equilateral triangle and the square are trivial, and Tabachnikov answered this for
Outer_billiards
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)
Roundly tapered end of a two- or three-dimensional object
ogive (/ˈoʊdʒaɪv/ OH-jyve) is the roundly tapered end of a two- or three-dimensional object. Ogive curves and surfaces are used in engineering, architecture
Ogive
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
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
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
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
Triangle whose side lengths and area are integers
are primitive. Since the area of an equilateral triangle with rational sides is an irrational number, no equilateral triangle is Heronian. However, a sequence
Heronian_triangle
Polygon with 18 edges
2{9/2} and 2{9/4} or two enneagrams, {18/6} is reduced to 6{3} or 6 equilateral triangles, and finally {18/9} is reduced to 9{2} as nine digons. Deeper
Octadecagon
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
Coordinate system that is defined by points instead of vectors
(a triangle for points in a plane, a tetrahedron for points in three-dimensional space, etc.). The barycentric coordinates of a point can be interpreted
Barycentric_coordinate_system
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
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
the same angle. Computing the maximum number of equiangular lines in n-dimensional Euclidean space is a difficult problem, and unsolved in general, though
Equiangular_lines
Space-filling polyhedron with 8 faces
figure. A topologically distinct elongated gyrobifastigium has square and equilateral triangle faces, seen as 2 triangular prisms augmented to a central cube
Elongated_gyrobifastigium
Compact Riemann surface of genus 3
each of degree 3 (meeting at 56 vertices), and the dual tiling by 56 equilateral triangles, each of degree 7 (meeting at 24 vertices). The order of the
Klein_quartic
Eigenvalue problem for the Laplace operator
century: the rectangular membrane by Siméon Denis Poisson in 1829, the equilateral triangle by Gabriel Lamé in 1852, and the circular membrane by Alfred
Helmholtz_equation
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
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
Polytope made by turning a polytope's facets into pyramids
and polyhedral combinatorics, the Kleetope of a polyhedron or higher-dimensional convex polytope P is another polyhedron or polytope PK formed by replacing
Kleetope
Construction for n-dimensional noise functions
Simplex noise is the result of an n-dimensional noise function comparable to Perlin noise ("classic" noise) but with fewer directional artifacts, in higher
Simplex_noise
Property of objects which are scaled or mirrored versions of each other
similar to each other, all squares are similar to each other, and all equilateral triangles are similar to each other. On the other hand, ellipses are
Similarity_(geometry)
Triangular prism capped with a tetrahedron
covers an equilateral triangle, replacing it with three other equilateral triangles, so that the resulting polyhedron has four equilateral triangles and
Elongated_triangular_pyramid
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