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An octant in
An octant in solid geometry
Solid geometry or stereometry is the geometry of Three-dimensional space, three-dimensional Euclidean space (3D space).
A solid figure is the region (mathematics), region of 3D space bounded by a two-dimensional closed surface; for example, a ...
is one of the eight divisions of a Euclidean three-dimensional coordinate system defined by the signs of the coordinates. It is analogous to the two-dimensional ''quadrant'' and the one-dimensional ''ray''.
The generalization of an octant is called ''orthant
In geometry, an orthant or hyperoctant is the analogue in ''n''-dimensional Euclidean space of a quadrant in the plane or an octant in three dimensions.
In general an orthant in ''n''-dimensions can be considered the intersection of ''n'' mutu ...
'' or ''hyperoctant''.
Naming and numbering
A convention for naming an octant is to give its list of signs, e.g. (+,−,−) or (−,+,−). Octant (+,+,+) is sometimes referred to as the ''first octant'', although similar ordinal name descriptors are not defined for the other seven octants. The advantages of using the (±,±,±) notation are its unambiguousness, and extensibility for higher dimensions. The following table shows the sign tuples together with likely ways to enumerate them. A binary enumeration with − as 1 can be easily generalized across dimensions. A binary enumeration with + as 1 defines the same order asbalanced ternary
Balanced ternary is a ternary numeral system (i.e. base 3 with three Numerical digit, digits) that uses a balanced signed-digit representation of the integers in which the digits have the values −1, 0, and 1. This stands in contrast to the stand ...
.
The Roman enumeration of the quadrants is in Gray code
The reflected binary code (RBC), also known as reflected binary (RB) or Gray code after Frank Gray (researcher), Frank Gray, is an ordering of the binary numeral system such that two successive values differ in only one bit (binary digit).
For ...
order, so the corresponding Gray code is also shown for the octants.
Verbal descriptions are ambiguous, because they depend on the representation of the coordinate system.
In the two depicted representations of a right-hand coordinate system, the first octant could be called ''right-back-top'' or ''right-top-front'' respectively.
See also
*Orthant
In geometry, an orthant or hyperoctant is the analogue in ''n''-dimensional Euclidean space of a quadrant in the plane or an octant in three dimensions.
In general an orthant in ''n''-dimensions can be considered the intersection of ''n'' mutu ...
* Octant (plane geometry)
* Octree
An octree is a tree data structure in which each internal node has exactly eight child node, children. Octrees are most often used to partition a three-dimensional space by recursive subdivision, recursively subdividing it into eight Octant (geo ...
* Spherical octant, the intersection of an octant of space and a sphere
* Trirectangular tetrahedron
References
{{reflist Euclidean solid geometry