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35-467: With late applications, 17,623 medals were awarded, including 15,300 to members of Canadian units. The medal, 1.4 inches (36 mm) in diameter , is silver and has a plain straight swivel suspender. The obverse bears the head of Queen Victoria with the legend VICTORIA REGINA ET IMPERATRIX , while the reverse depicts the ensign of Canada surrounded by a wreath of maple leaves with the word CANADA above. The recipient's name, rank and unit appear on

70-523: A code point in Unicode at U+2300 ⌀ DIAMETER SIGN , in the Miscellaneous Technical set. It should not be confused with several other characters (such as U+00D8 Ø LATIN CAPITAL LETTER O WITH STROKE or U+2205 ∅ EMPTY SET ) that resemble it but have unrelated meanings. It has the compose sequence Compose d i . The diameter of

105-418: A diameter of a circle is any straight line segment that passes through the centre of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid for the diameter of a sphere . In more modern usage, the length d {\displaystyle d} of a diameter is also called the diameter. In this sense one speaks of

140-669: A hypercube or a set of scattered points. The diameter or metric diameter of a subset of a metric space is the least upper bound of the set of all distances between pairs of points in the subset. Explicitly, if S {\displaystyle S} is the subset and if ρ {\displaystyle \rho } is the metric , the diameter is diam ⁡ ( S ) = sup x , y ∈ S ρ ( x , y ) . {\displaystyle \operatorname {diam} (S)=\sup _{x,y\in S}\rho (x,y).} If

175-399: A ( 1 {\displaystyle 1} -dimensional) line segment has 2 {\displaystyle 2} endpoints; a ( 2 {\displaystyle 2} -dimensional) square has 4 {\displaystyle 4} sides or edges; a 3 {\displaystyle 3} -dimensional cube has 6 {\displaystyle 6} square faces;

210-517: A ( 4 {\displaystyle 4} -dimensional) tesseract has 8 {\displaystyle 8} three-dimensional cubes as its facets. The number of vertices of a hypercube of dimension n {\displaystyle n} is 2 n {\displaystyle 2^{n}} (a usual, 3 {\displaystyle 3} -dimensional cube has 2 3 = 8 {\displaystyle 2^{3}=8} vertices, for instance). The number of

245-458: A circle is exactly twice its radius. However, this is true only for a circle, and only in the Euclidean metric . Jung's theorem provides more general inequalities relating the diameter to the radius. Hypercube In geometry , a hypercube is an n -dimensional analogue of a square ( n = 2 ) and a cube ( n = 3 ); the special case for n = 4 is known as a tesseract . It

280-477: A simple combinatorial argument: for each of the 2 n {\displaystyle 2^{n}} vertices of the hypercube, there are ( n m ) {\displaystyle {\tbinom {n}{m}}} ways to choose a collection of m {\displaystyle m} edges incident to that vertex. Each of these collections defines one of the m {\displaystyle m} -dimensional faces incident to

315-515: Is 2 {\displaystyle 2} , and its n {\displaystyle n} -dimensional volume is 2 n {\displaystyle 2^{n}} . Every hypercube admits, as its faces, hypercubes of a lower dimension contained in its boundary. A hypercube of dimension n {\displaystyle n} admits 2 n {\displaystyle 2n} facets, or faces of dimension n − 1 {\displaystyle n-1} :

350-409: Is p vertices and pn facets. Any positive integer raised to another positive integer power will yield a third integer, with this third integer being a specific type of figurate number corresponding to an n -cube with a number of dimensions corresponding to the exponential. For example, the exponent 2 will yield a square number or "perfect square", which can be arranged into a square shape with

385-528: Is a closed , compact , convex figure whose 1- skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions , perpendicular to each other and of the same length. A unit hypercube's longest diagonal in n dimensions is equal to n {\displaystyle {\sqrt {n}}} . An n -dimensional hypercube is more commonly referred to as an n -cube or sometimes as an n -dimensional cube . The term measure polytope (originally from Elte, 1912)

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420-459: Is also used, notably in the work of H. S. M. Coxeter who also labels the hypercubes the γ n polytopes. The hypercube is the special case of a hyperrectangle (also called an n-orthotope ). A unit hypercube is a hypercube whose side has length one unit . Often, the hypercube whose corners (or vertices ) are the 2 points in R with each coordinate equal to 0 or 1 is called the unit hypercube. A hypercube can be defined by increasing

455-461: Is any chord passing through the centre of the ellipse. For example, conjugate diameters have the property that a tangent line to the ellipse at the endpoint of one diameter is parallel to the conjugate diameter. The longest diameter is called the major axis . The word "diameter" is derived from Ancient Greek : διάμετρος ( diametros ), "diameter of a circle", from διά ( dia ), "across, through" and μέτρον ( metron ), "measure". It

490-642: Is either equal to 1 / 2 {\displaystyle 1/2} or to − 1 / 2 {\displaystyle -1/2} . This unit hypercube is also the cartesian product [ − 1 / 2 , 1 / 2 ] n {\displaystyle [-1/2,1/2]^{n}} . Any unit hypercube has an edge length of 1 {\displaystyle 1} and an n {\displaystyle n} -dimensional volume of 1 {\displaystyle 1} . The n {\displaystyle n} -dimensional hypercube obtained as

525-477: Is often abbreviated DIA , dia , d , {\displaystyle {\text{DIA}},{\text{dia}},d,} or ∅ . {\displaystyle \varnothing .} The definitions given above are only valid for circles, spheres and convex shapes. However, they are special cases of a more general definition that is valid for any kind of n {\displaystyle n} -dimensional (convex or non-convex) object, such as

560-489: Is one of three regular polytope families, labeled by Coxeter as γ n . The other two are the hypercube dual family, the cross-polytopes , labeled as β n, and the simplices , labeled as α n . A fourth family, the infinite tessellations of hypercubes , is labeled as δ n . Another related family of semiregular and uniform polytopes is the demihypercubes , which are constructed from hypercubes with alternate vertices deleted and simplex facets added in

595-546: Is the convex hull of all the 2 n {\displaystyle 2^{n}} points whose n {\displaystyle n} Cartesian coordinates are each equal to either 0 {\displaystyle 0} or 1 {\displaystyle 1} . These points are its vertices . The hypercube with these coordinates is also the cartesian product [ 0 , 1 ] n {\displaystyle [0,1]^{n}} of n {\displaystyle n} copies of

630-581: Is the length of the edges of the hypercube. These numbers can also be generated by the linear recurrence relation . For example, extending a square via its 4 vertices adds one extra line segment (edge) per vertex. Adding the opposite square to form a cube provides E 1 , 3 = 12 {\displaystyle E_{1,3}=12} line segments. The extended f-vector for an n -cube can also be computed by expanding ( 2 x + 1 ) n {\displaystyle (2x+1)^{n}} (concisely, (2,1) ), and reading off

665-446: Is the same as the diameter of its convex hull . In medical terminology concerning a lesion or in geology concerning a rock, the diameter of an object is the least upper bound of the set of all distances between pairs of points in the object. In differential geometry , the diameter is an important global Riemannian invariant . In planar geometry , a diameter of a conic section is typically defined as any chord which passes through

700-879: The m {\displaystyle m} -dimensional hypercubes (just referred to as m {\displaystyle m} -cubes from here on) contained in the boundary of an n {\displaystyle n} -cube is For example, the boundary of a 4 {\displaystyle 4} -cube ( n = 4 {\displaystyle n=4} ) contains 8 {\displaystyle 8} cubes ( 3 {\displaystyle 3} -cubes), 24 {\displaystyle 24} squares ( 2 {\displaystyle 2} -cubes), 32 {\displaystyle 32} line segments ( 1 {\displaystyle 1} -cubes) and 16 {\displaystyle 16} vertices ( 0 {\displaystyle 0} -cubes). This identity can be proven by

735-449: The width is often defined to be the smallest such distance. Both quantities can be calculated efficiently using rotating calipers . For a curve of constant width such as the Reuleaux triangle , the width and diameter are the same because all such pairs of parallel tangent lines have the same distance. For an ellipse , the standard terminology is different. A diameter of an ellipse

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770-478: The conic's centre ; such diameters are not necessarily of uniform length, except in the case of the circle, which has eccentricity e = 0. {\displaystyle e=0.} The symbol or variable for diameter, ⌀ , is sometimes used in technical drawings or specifications as a prefix or suffix for a number (e.g. "⌀ 55 mm"), indicating that it represents diameter. Photographic filter thread sizes are often denoted in this way. The symbol has

805-409: The diameter rather than a diameter (which refers to the line segment itself), because all diameters of a circle or sphere have the same length, this being twice the radius r . {\displaystyle r.} For a convex shape in the plane , the diameter is defined to be the largest distance that can be formed between two opposite parallel lines tangent to its boundary, and

840-533: The vertex figure are regular simplexes . The regular polygon perimeter seen in these orthogonal projections is called a Petrie polygon . The generalized squares ( n = 2) are shown with edges outlined as red and blue alternating color p -edges, while the higher n -cubes are drawn with black outlined p -edges. The number of m -face elements in a p -generalized n -cube are: p n − m ( n m ) {\displaystyle p^{n-m}{n \choose m}} . This

875-433: The coefficients of the resulting polynomial . For example, the elements of a tesseract is (2,1) = (4,4,1) = (16,32,24,8,1). An n -cube can be projected inside a regular 2 n -gonal polygon by a skew orthogonal projection , shown here from the line segment to the 16-cube. The hypercubes are one of the few families of regular polytopes that are represented in any number of dimensions. The hypercube (offset) family

910-465: The considered vertex. Doing this for all the vertices of the hypercube, each of the m {\displaystyle m} -dimensional faces of the hypercube is counted 2 m {\displaystyle 2^{m}} times since it has that many vertices, and we need to divide 2 n ( n m ) {\displaystyle 2^{n}{\tbinom {n}{m}}} by this number. The number of facets of

945-492: The convex hull of the points with coordinates ( ± 1 , ± 1 , ⋯ , ± 1 ) {\displaystyle (\pm 1,\pm 1,\cdots ,\pm 1)} or, equivalently as the Cartesian product [ − 1 , 1 ] n {\displaystyle [-1,1]^{n}} is also often considered due to the simpler form of its vertex coordinates. Its edge length

980-414: The empty set as a special case, assigning it a diameter of 0 , {\displaystyle 0,} which corresponds to taking the codomain of ρ {\displaystyle \rho } to be the set of nonnegative reals. For any solid object or set of scattered points in n {\displaystyle n} -dimensional Euclidean space , the diameter of the object or set

1015-498: The gaps, labeled as hγ n . n -cubes can be combined with their duals (the cross-polytopes ) to form compound polytopes: The graph of the n -hypercube's edges is isomorphic to the Hasse diagram of the ( n −1)- simplex 's face lattice . This can be seen by orienting the n -hypercube so that two opposite vertices lie vertically, corresponding to the ( n −1)-simplex itself and the null polytope, respectively. Each vertex connected to

1050-492: The hypercube can be used to compute the ( n − 1 ) {\displaystyle (n-1)} -dimensional volume of its boundary: that volume is 2 n {\displaystyle 2n} times the volume of a ( n − 1 ) {\displaystyle (n-1)} -dimensional hypercube; that is, 2 n s n − 1 {\displaystyle 2ns^{n-1}} where s {\displaystyle s}

1085-469: The metric ρ {\displaystyle \rho } is viewed here as having codomain R {\displaystyle \mathbb {R} } (the set of all real numbers ), this implies that the diameter of the empty set (the case S = ∅ {\displaystyle S=\varnothing } ) equals − ∞ {\displaystyle -\infty } ( negative infinity ). Some authors prefer to treat

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1120-517: The numbers of dimensions of a shape: This can be generalized to any number of dimensions. This process of sweeping out volumes can be formalized mathematically as a Minkowski sum : the d -dimensional hypercube is the Minkowski sum of d mutually perpendicular unit-length line segments, and is therefore an example of a zonotope . The 1- skeleton of a hypercube is a hypercube graph . A unit hypercube of dimension n {\displaystyle n}

1155-584: The rim of the medal. A number of different impressed and engraved styles were used, reflecting that the medal was awarded over a long period of time. The 1.25 inches (32 mm) wide ribbon consists of three equal stripes of red, white and red. In 1943 the same ribbon was adopted for the Canada Medal . The medal was always awarded with a clasp, with 12 medals awarded with all three clasps. The number of clasps indicated below includes those that appear on multi-clasp medals. Diameter In geometry ,

1190-1044: The top vertex then uniquely maps to one of the ( n −1)-simplex's facets ( n −2 faces), and each vertex connected to those vertices maps to one of the simplex's n −3 faces, and so forth, and the vertices connected to the bottom vertex map to the simplex's vertices. This relation may be used to generate the face lattice of an ( n −1)-simplex efficiently, since face lattice enumeration algorithms applicable to general polytopes are more computationally expensive. Regular complex polytopes can be defined in complex Hilbert space called generalized hypercubes , γ n = p {4} 2 {3}... 2 {3} 2 , or [REDACTED] [REDACTED] [REDACTED] [REDACTED] .. [REDACTED] [REDACTED] [REDACTED] [REDACTED] . Real solutions exist with p = 2, i.e. γ n = γ n = 2 {4} 2 {3}... 2 {3} 2 = {4,3,..,3}. For p > 2, they exist in C n {\displaystyle \mathbb {C} ^{n}} . The facets are generalized ( n −1)-cube and

1225-453: The unit interval [ 0 , 1 ] {\displaystyle [0,1]} . Another unit hypercube, centered at the origin of the ambient space, can be obtained from this one by a translation . It is the convex hull of the 2 n {\displaystyle 2^{n}} points whose vectors of Cartesian coordinates are Here the symbol ± {\displaystyle \pm } means that each coordinate

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