Hypercube Software LINK Download
Angelines Mulready
In geometry, a hypercube is an n-dimensional analogue of a square (n = 2) and a cube (n = 3). It 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.[citation needed] The term measure polytope (originally from Elte, 1912)[1] is also used, notably in the work of H. S. M. Coxeter who also labels the hypercubes the γn polytopes.[2]
A unit hypercube is a hypercube whose side has length one unit. Often, the hypercube whose corners (or vertices) are the 2n points in Rn with each coordinate equal to 0 or 1 is called the unit hypercube.
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.
A unit hypercube of dimension n \displaystyle n is the convex hull of all the points whose n \displaystyle n Cartesian coordinates are each equal to either 0 \displaystyle 0 or 1 \displaystyle 1 . This hypercube is also the cartesian product [ 0 , 1 ] n \displaystyle [0,1]^n of n \displaystyle n copies of 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 points whose vectors of Cartesian coordinates are
The hypercube (offset) family is one of three regular polytope families, labeled by Coxeter as γn. The other two are the hypercube dual family, the cross-polytopes, labeled as βn, and the simplices, labeled as αn. A fourth family, the infinite tessellations of hypercubes, he labeled as δn.
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 the gaps, labeled as hγn.
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 a side length corresponding to that of the base. Similarly, the exponent 3 will yield a perfect cube, an integer which can be arranged into a cube shape with a side length of the base. As a result, the act of raising a number to 2 or 3 is more commonly referred to as "squaring" and "cubing", respectively. However, the names of higher-order hypercubes do not appear to be in common use for higher powers.
I am trying to visualise hypercube like the below. I have asked help here but getting nodes on top of each other with the packages there. The N-th hypercube has 2^n nodes and each vertex with the degree of n. I am trying to find a way to vizualise hypercubes with larger degree so big challenge for the pkg. My goal is to vizualise traversing of the cube. How would you do this kind of vizualisation?
I would like to use an n-dimensional hypercube to do so for convenience. Using MPI_Cart_Create I can create self-organising dimensions. Doing so will maximize efficiency of my process and also reduce the number of LOC I have to spit to get it done..
so that would be for a 4-d hypercube.. The pattern is pretty straight-forward. In n-dimensional hypercube each point have N neighbour and they are represented in this code. Note that this code should used instead of xoring bit mask because MPI can re-order the processes to fit the physical layout of your clusters.
A template taxonomy imports all domain member taxonomies and primary taxonomies and adds the dimensional structures that will be used in the XBRL instance. By convention, a taxonomy that imports primary and domain member taxonomies and defines all the necessary dimensional information is called a template taxonomy. In particular, a template defines Hypercube. A Hypercube describes the Cartesian product of zero or more dimensions. Each dimension, in turn, is defined over zero or more domains and domains are composed of members. Note that in this formulation, a hypercube of a Primary Item does not include the Primary Item itself.
[Def, 1] Primary item declarations are elements defined in XBRL taxonomies that are in the xbrli:item substitution group and are not in the xbrldt:hypercubeItem or xbrldt:dimensionItem substitution group.
[Def, 4] A hypercube declaration is an abstract item declaration in the xbrldt:hypercubeItem substitution group. A hypercube is an ordered list of dimensions, defined by the set of zero or more dimension declarations linked to the hypercube by hypercube-dimension relationships in a dimensional relationship set [Def, 3], and ordered according to the @order attribute of these relationships.
The order of the hypercube-dimension relationship for taxonomy representation purposes in taxonomy editing tools is defined by the value of the @order attribute on the arc defining the relationship.
Example 1 shows a hypercube consisting of two typed dimensions - , Team and Drink. This example shows a hypercube describing the occurrence of Team and Drink elements in either the or element of a .
A set of hypercubes MAY be composed via conjunction of all and notAll compositors. The relationship between a compositor and its operands is represented by [XLINK] arcs with distinct arc roles to define the different operators.
These relationships MAY be in different base sets. When has-hypercube relationships are in different base sets, a Primary Item that is dimensionally valid in any base set is dimensionally valid.
Relationships in the dimensional relationship set [Def, 3] of an relationship are relevant to instance validation. The source and target are primary item declarations and hypercube declarations [Def, 4], respectively.
The instantiation of a primary item declaration [Def, 1] in an instance document is dimensionally valid with respect to a conjunction of hypercubes only if it is valid with respect to all of the conjoined hypercubes individually. A negated hypercube notAll is valid if the non negated version of the same hypercube definition is invalid. The conjunction of a single hypercube is the hypercube itself Example 2.
The primary item declaration p_FluidCapacity is associated with the composition of two hypercubes in the same base set. A will be valid with respect to the Primary Item only if it has a City reference in its that is a member of the hc_CityHypercubeAll and not a member of hc_CityHypercubeExcluded.
If the @xbrldt:closed attribute is specified with a true value on a has-hypercube arc with the value for the @xbrldt:contextElement attribute, the hypercube is closed with respect to the element in that base set.
If the @xbrldt:closed attribute is specified with a true value on a has-hypercube arc with the value scenario for the @xbrldt:contextElement attribute, the hypercube is closed with respect to the element in that base set.
[Def, 21] The instantiation of a primary item declaration [Def, 1] as a fact in an instance document is dimensionally valid witih respect to a closed hypercube when no other elements are children of a or element except those appearing in the closed hypercube. dimensionally valid with respect to a closed hypercube when:
The arcs with xbrldt:closed="true" mean that a is valid with respect to the target hypercube if it has a Team and Drink and nothing else in the element, and nothing at all in the element. Note that the all arc to hc_Team_x_Drink has segment in its @xbrldt:contextElement attribute and the all arc to hc_Empty has in its @xbrldt:contextElement attribute.
Furthermore, a set of primary item declarations MAY have hypercubes in common among the targets of their has-hypercube relationships.;hHypercube declarations in turn MAY have typed dimensions in common among the targets of their hypercube-dimension relationships. In Section 2.5.2 and Section 2.5.3, additional relationships will also introduce tangled graphs, with some items as the source of separate and distinct sets of relationships to define different dimensions. If all the dimensional relationships used together in a validation were forced to be in the same base set, there would be redundancy among dimensional relationships, violating Requirement P4.
The optional @xbrldt:targetRole attribute on an arc allows a taxonomy author to connect together two arcs that represent a consecutive relationship [Def, 2] that exist in different rolesbase sets. As declared in this document, the @xbrldt:targetRole attribute MAY appear on definition arcs having the following arc roles: all, notAll, hypercube-dimension, dimension-domain and domain-member. The @xbrldt:targetRole attribute has the type anyURI. Resolution of the URI is not subject to the presence of an @xml:base attribute and its value MUST be an absolute URI.
Two arcs that represent a consecutive relationship [Def, 2] that exist in different extended-type link elements MUST be connected together using the @xbrldt:targetRole attribute. Not doing so causes the construction to be unconnected and results in, for example, empty hypercubes, dimensions and domains.
A primary item declaration MAY be the source of a domain-member relationship. When a primary item declaration is the source of both a domain-member relationship and a has-hypercube relationship, the target of the domain-member arc relationship is said to inherit the has-hypercube relationship of the source element. Inheritance is transitive. Inheritance preserves the base set and DRS of the original has-hypercube relationship, as well as the values of its @xbrldt:contextElement attribute and @xbrldt:closed attribute.
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