# Multi-dimensional Glossary

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## Last revised 2003-12-08

guide: word part of speech [author or source of word] - definition

ana adverb [greek: "up"] - A term used in some literature to refer to one of the two directions from three dimensional space into four dimensional space. In tetraspace, represented by the vector <0,0,0,1>. This term refers to the direction analogous to up; the opposite direction is kata. On this webpage I use the term upsilon for the concept of four-dimensional up. See the chart under direction.

apos noun [Oren] - The portion of an object that is furthest ana. It is the analog of the top portion of an object for the direction up, or the front portion for the forward direction. Derived from "apex". The opposite side of an object is its zakos.

area noun - The measure of space within a two dimensional object. It is the planespace analog of volume in realmspace and bulk in tetraspace.

backward adverb - The direction towards the space behind an object; movement in the negative z direction. In realmspace, represented by the vector <0,0,-1>. The opposite direction is forward. See the chart under direction.

ball noun - A sphere and all of its inside points. Defined as all points <= radius r away from a center point, in three dimensions. In mathematics, this term refers to the interior of an n-dimensional sphere. It is the realmspace analog of the disk in planespace and gongyl in tetraspace. See the chart under rotatope.

bionian noun [Aale de Winkel] - A person from planespace. adjective - From planespace. For example, a "bionian river" is a river from planespace. See chart under tetraspace.

bulk noun [George Olshevsky] - The measure of space within a four dimensional object. It is the tetraspace analog of area in planespace and volume in realmspace.

cardinal direction noun - A direction toward or away from either planetary rotation or the planetary poles. Bionian planets only have east and west; trionian planets additionally have north and south; tetronian planets have two more, marp and garp

cell noun - A three-dimensional unit of a four-or-higher dimensional figure. A cell is the intersection point of two terons; two cells meet at a face. It is the realmic analog of vertex, edge, face, and teron

chasm noun - A planar division between two sets of points; a gap that separates two objects in realmspace or lower. A chasm can penetrate objects in tetraspace or higher without splitting them. It is the realmspace analog of the hole in planespace and the gully in tetraspace.

circle noun - Two-dimensional hypersphere. Defined as all points a distance r away from a center point, in two dimensions. It is the planespace analog of the sphere in realmspace and glome in tetraspace. Its solid version is the disk. See the chart under rotatope

column noun - A sequence of objects or data arranged in a line vertically. A horizontal sequence is a row and a depthwise sequence is a pillar.

cone noun - A shape formed by spinning an isosceles triangle into realmspace, around its symmetry axis. Can also be formed by connecting all points on the edge of a circle to a point some distance away from the circle in realmspace, up or down from the center of the circle. It is the analog of the sphone in tetraspace.

crind noun - A 3d object that is the intersection of two perpendicular cylinders. It has two linear axes of rotation like the duocylinder, but it is not a rotatope because the angles between the sides aren't right angles.

cube noun - Three-dimensional hypercube; its vertices are (+/-1, +/-1, +/-1). It is a polyhedron formed from six equal square faces perpendicular to each other at their edges. It is the realmspace analog of the square in planespace and the tesseract in tetraspace.

cubinder noun [Garrett Jones] - A four dimensional hypercylinder that can roll along one axis, but is flat on all the other sides. It can be formed by extending a cylinder into tetraspace linearly. See the chart under rotatope

cylinder noun - A three dimensional hypercylinder that can roll along one linear axis. It can be formed by extending a circle into realmspace, or by rotating a square into realmspace. See the chart under rotatope

delta adverb - One of the two directions pointing out of realmspace into tetraspace; movement in the negative w direction. In tetraspace, represented by the vector <0,0,0,-1>. This term refers to the direction analogous to down; the 'd' in 'delta' corresponds to the 'd' in 'down'. In some literature, this direction is referred to as ana. This term is the same direction as zant, but is used in a different context: delta is movement from realmspace to tetraspace, but zant is a turning direction.The opposite direction is upsilon. See the chart under direction.

direction noun - A straight path away from an object that relative to the axes of an object. Directions on a planet are cardinal directions

 axis vector - direction + direction x <+/-1,0,0,0> left right y <0,+/-1,0,0> down up z <0,0,+/-1,0> backward forward w <0,0,0,+/-1> kata, delta, zant ana, upsilon, wint

disk noun - A circle and all of its inside points. Defined as all points <= radius r away from a center point, in two dimensions. It is the planespace analog of the ball in realmspace and the gongyl in tetraspace. See the chart under rotatope

double rotation noun - Two independent rotations on a 4 or higher dimensional object. When both rotations are of the same magnitude, the double rotation is called a swirl.

down adverb - The direction towards gravity; movement in the negative y direction. In realmspace, represented by the vector <0,-1,0>. The opposite direction is up. See the chart under direction.

duocylinder noun [Jonathan Bowers] - A four dimensional shape that can roll along two independent linear axes. If it is placed on one rolling axis, it cannot roll along the other axis, but it can roll no matter which side it is placed on. It can be formed by rotating the flat ends of a cylinder into tetraspace. See the chart under rotatope

east noun - The cardinal direction towards planetary rotation. The opposite direction is west. Distance east or west is measured in longitude

edge noun - A one-dimensional unit of a two-or-higher dimensional figure. An edge is the intersection point of two faces; two edges meet at a vertex. It is the linear analog of vertex, face, cell, and teron

equator noun - The largest circle on the surface of a planet that is parallel to the rotation of the planet; equidistant from the poles and perpendicular to the axis of rotation. A planet in realmspace has a single equator; a planet in tetraspace has a solar equator and a polar equator

extrude verb - To create a new shape by tracing the path of the original shape in a direction perpendicular to it.

face noun - A two-dimensional unit of a three-or-higher dimensional figure. A face is the intersection point of two cells; two faces meet at an edge. It is the planar analog of vertex, edge, face, and teron

flune noun [Garrett Jones] - A 4d hyperplane; an infinite set of points extending in four dimensions. It is the 4d analog of the line (1d), plane (2d), and realm (3d). Also called a tetrealm. See the chart under tetraspace.

forward adverb - The direction towards the space in front of an object; movement in the positive z direction. In realmspace, represented by the vector <0,0,1>. The opposite direction is backward. See the chart under direction.

garp noun [Jonathan Bowers] - The cardinal direction in the opposite direction of polar rotation (rotation along the polar equator). The opposite direction is marp. Distance towards marp or garp is measured in laptitude

glome noun [George Olshevsky] - Four-dimensional hypersphere. Defined as all points a distance r away from a center point, in four dimensions. It is the tetraspace analog of the circle in planespace and the sphere in realmspace. Its solid version is the gongyl. See the chart under rotatope

gongyl noun [George Olshevsky] - A glome and all of its inside points. Defined as all points <= radius r away from a center point, in four dimensions. It is the tetraspace analog of the disk in planespace and ball in realmspace. See the chart under rotatope.

gully noun - A realmic division between two sets of points; a gap that separates two objects in tetraspace or lower. A gully can penetrate objects in pentaspace or higher without splitting them. It is the tetraspace analog of the hole in planespace and the chasm in realmspace.

hole noun - A linear division between two sets of points; a gap that separates two objects in planespace or lower. A hole can penetrate objects in realmspace or higher without splitting them. It is the planespace analog of the chasm in realmspace and the gully in teteraspace.

hypercube noun - A shape of n dimensions where the vertices in each dimension are +/-1. It is the n-dimensional analog of the square in planespace, the cube in realmspace, and the tesseract in tetraspace. All of a hypercube's faces are (n-1) dimensional hypercubes. Sometimes the term is restricted only to dimensions above the third, and frequently it is used to refer only to the tesseract.

hypercylinder noun - An n-dimensional cylinder. Hypercylinders are formed by extending or rotating lower-dimensional hypercylinders into higher dimensional space. In realmspace there is only one hypercylinder, the cylinder. In tetraspace, there are two hypercylinders: the cubinder and spherinder

hyperplane noun - An infinite set of points extending into n dimensions. A one dimensional hyperplane is a line, two-dimensional is a plane, three dimensional is a realm, and four-dimensional is a flune. A hyperplane of n dimensions intersects a plane of n+1 dimensions into two parts. See the chart under tetraspace.

hyperspace noun - Space above the third dimension. Tetraspace is the fourth dimension.

hypersphere noun - The set of points a distance r away from a center point in n dimensions. It is the n-dimensional analog of the circle in planespace, the sphere in realmspace, and the glome in tetraspace. Sometimes the term hypersphere is used to refer only to the glome. See the chart under rotatope

kata adverb [greek: "down"] - a term used in some literature to refer to one of the two directions from three dimensional space into four dimensional space. In tetraspace, represented by the vector <0,0,0,-1>. This term refers to the direction analogous to down; the opposite direction is ana. On this webpage I use the term delta for the concept of four-dimensional down. See the chart under direction.

laptitude noun [Jonathan Bowers] - Angular distance on the surface of a tetronian planet that measures distance marp/garp from the solar equator. Each laptitude is in the shape of a hemisphere.

latitude noun - Angular distance of the surface of a planet that measures distance north/south from the equator. Latitudinal lines on a trionian planet run east/west, and are in the shape of circles. Each latitude on tetronian planet is in the shape of a torus.

laver noun [Jonathan Bowers] - A planar body of water in tetraspace or higher, which may have currents flowing in various directions. (laver = lake + river)

left adverb - Opposite of right; movement in the negative x direction. In realmspace, represented by the vector <-1,0,0>. See right for the description of the turning directions in realmspace. See the chart under direction.

line noun - A one dimensional hyperplane; an infinite set of points extending in one dimension. It is the 1d analog of the plane (2d), realm (3d), and flune (4d). A line intersects a plane into two parts. See the chart under tetraspace.

linear adj - Long in one dimension, thin in every other dimension. Examples of linear objects are pencils, rivers, poles, strings, etc. See the chart under tetraspace.

linespace noun - One dimensional space; consists of a single line. See the chart under tetraspace.

longitude noun - Angular distance on the surface of a planet that measures distance east/west from a prime meridian. Longitudinal lines on a trionian planet run north/south, and are in the shape of half-circles. Each longitude on a tetronian planet is in the shape of a hemisphere.

marp noun [Jonathan Bowers] - The cardinal direction in the direction of polar rotation (rotation along the polar equator). The opposite direction is garp. Distance towards marp or garp is measured in laptitude

multiverse noun [Mark Gruenwald] - A set of universes.

north noun - One of the cardinal directions towards a planetary pole; in realmspace, 90 degrees counter-clockwise from east; in tetraspace, the direction towards the polar equator. The opposite direction is south. Distance towards north or south is measured in latitude

omniverse noun [Mark Gruenwald] - The set of all universes in existence.

pentaspace noun - Five-dimensional space. See the chart under tetraspace.

perimeter noun - The boundary of a two-dimensional shape, measured in length. It is the analog of surface in realmspace and surcell in tetraspace.

pillar noun - A sequence of objects or data arranged depthwise (as opposed to vertically or horizontally). A horizontal sequence is a row, and a vertical sequence is a column.

planar adj - Wide in two dimensions, thin in every other dimension. Examples of planar objects are circles, squares, sheets of paper, cd's, the surface of water, etc. See the chart under tetraspace.

plane noun - A two dimensional hyperplane; an infinite set of points extending in two dimensions. It is the 2d analog of the line (1d), realm (3d), and flune (4d). A plane intersects a realm into two parts. See the chart under tetraspace.

planespace noun - Two dimensional space; consists of a single plane. See the chart under tetraspace.

polar equator noun - Equator on a planet in tetraspace that is orthogonal to the solar equator; rotates around the solar equator.

pole noun - A point on a planet where the axis of rotation and the surface of the planet intersect. A planet in realmspace has two poles, but a planet in tetraspace has two equators instead - each equator functions as the other equator's pole.

polychoron noun - A four-dimensional shape; bounded by polyhedra. An example of a polytope.

polygon noun - A two-dimensional shape; bounded by line segments. An example of a polytope.

polyhedron noun - A three-dimensional solid; bounded by polygons. An example of a polytope.

polyteron noun - A five-dimensional shape; bounded by polychora. An example of a polytope.

polytope noun - An n-dimensional solid, bounded by (n-1) dimensional solids. This includes polygons in planespace, polyhedra in realmspace, polychora in tetraspace, and polytera in pentaspace.

realm noun [George Olshevsky] - A three dimensional hyperplane; an infinite set of points extending in three dimensions. It is the 3d analog of the line (1d), plane (2d), and flune (4d). A realm intersects flune into two parts. See the chart under tetraspace.

realmic adj - Thick in three dimensions, thin in every other dimension. Examples of realmic objects are cubes, spheres, chairs, humans, gerbils, microwaves, and all other three-dimensional objects that aren't linear or planar.See the chart under tetraspace.

realmspace noun - Three dimensional space; consists of a single realm. It is the space that humans exist in. See the chart under tetraspace.

right adverb - One of the two turning directions of a trionian object; movement in the positive x direction. In realmspace, represented by the vector <1,0,0>. If one looks at the movement directions from a viewpoint above and traverses them in a clockwise direction starting at left, the order is left, forward, right, backward. Thus, right is 90 degrees clockwise from forward and 90 degrees counter-clockwise from backward. Left is the opposite: 90 degrees counter-clockwise from forward and 90 degrees clockwise from backward. There are two extra turning directions in tetraspace, zant and wint. See the chart under direction.

rotachoron noun [Garrett Jones] - A four dimensional shape formed by extensions and rotations. This includes the tesseract, cubinder, duocylinder, spherinder, and glome. See the chart under rotatope.

rotagon noun [Garrett Jones] - A two dimensional shape formed by extensions and rotations. This includes the square and circle. See the chart under rotatope.

rotahedron noun [Garrett Jones] - A three dimensional shape formed by extensions and rotations. This includes the cube, cylinder, and sphere. See the chart under rotatope.

rotatope noun [Garrett Jones] - An n-dimensional shape formed by extensions and rotations.

 dimension n-type n-cube n-cylinders m-sphere crosses n-sphere n-ball 2nd rotagon square - - circle disk 3rd rotahedron cube cylinder - sphere ball 4th rotachoron tesseract cubinder, spherinder duocylinder glome gongyl

row noun - A sequence of objects or data arranged in a line horizontally. A vertical sequence is a column and a depthwise sequence is a pillar

sheet noun - Any flexible planar object; can only exist in realmspace or higher. Trionians write on sheets of paper; tetronians can tie sheets into knots. It is the analog of the string in planespace and the swock in tetraspace.

skring noun [Jonathan Bowers] - A tetronian spring, constructed from a sheet. (skring = square + spring)

solar equator noun - The equator of a planet in tetraspace aligned most closely to the plane of orbit around the planet's star. The solar equator rotates around the polar equator.

south noun - One of the cardinal directions towards a planetary pole; in realmspace, 90 degrees clockwise from east; in tetraspace, the direction towards the solar equator. The opposite direction is north. Distance towards north or south is measured in latitude

sphere noun - Three-dimensional hypersphere. Defined as all points a distance r away from a center point, in three dimensions. It is the realmspace analog of the circle in planespace and the glome in tetraspace. Its solid version is the ball. See the chart under rotatope

spherinder noun [Jonathan Bowers] - A four dimensional hypercylinder that can roll around in a planar region. It can be formed by extending a sphere into tetraspace linearly. See the chart under rotatope

sphone noun [Garrett Jones] - A shape formed by spinning a cone into tetraspace, around its symmetry axis. Can also be formed by connected all the points on the surface of a sphere to a point some distance away from the sphere in tetraspace, with the point being upsilon or delta from the center of the sphere. It is the analog of the cone in realmspace. (sphone = sphere + cone).

square noun - Two-dimensional hypercube; its vertices are (+/-1, +/-1). It is a polygon formed from four equal line segments that are perpendicular to each other at their vertices. It is the planespace analog to the cube in realmspace and the tesseract in tetraspace.

string noun - Any flexible linear object; can only exist in planespace or higher. Bionians write on strings of paper; trionians can tie strings into knots; in tetraspace, one string cannot block the path of another string. It is the analog of the sheet in realmspace and the swock in tetraspace.

surcell noun [George Olshevsky] - The boundary of a four-dimensional shape, measured in volume. It is the analog of perimeter in planespace and surface in realmspace. (surcell = surface + cell)

surface noun - The boundary of a three-dimensional shape, measured in area. It is the analog of perimeter in planespace and surcell in tetraspace.

swirl noun [George Olshevsky] - A double rotation in hyperspace in which the magnitudes of the two rotations are equal. In a swirl, every point on the object traces a circle around the center of the rotations. There are left-handed and right-handed swirls, much as there are clockwise and counterclockwise rotations.

swock noun [Jonathan Bowers] - Any flexible realmic object; can only exist in tetraspace or higher. Tetronians write on swocks of paper; pentonians can tie swocks into knots. It is the analog of the string in planespace and the sheet in realmspace.

tarrow adj [Garrett Jones] - the four dimensional analog of short (length), narrow, and short (height). It is a measure of small trength. Its opposite is trong

teron noun [Wendy Kreiger] - A four-dimensional unit of a five-or-higher dimensional figure. Two terons meet at a cell. It is the tetrealmic analog of vertex, edge, face, and cell.

tesseract noun - Four-dimensional hypercube; its vertices are (+/-1, +/-1, +/-1, +/-1). It is a polychoron formed from eight equal cube cells that are perpendicular to each other at their faces. It is the tetraspace analog to the square in planespace and the cube in realmspace. In some literature, the term tesseract refers to the inner projection of a tetracube into realmspace, which looks like a cube within a cube.

tesserobject noun [George Olshevsky] - Any object in tetraspace, not necessarily a polytope.

tetral adj [Garrett Jones] - Wide in four dimensions, thin in all dimensions above four. Any object in tetraspace that isn't linear, planar, or realmic is tetral. See the chart under tetraspace.

tetraspace noun - Four dimensional space; consists of a single flune.

 dimension n-space n-plane adjective 1st linespace line linear 2nd planespace plane planar 3rd realmspace realm realmic 4th tetraspace flune, tetrealm tetral, tetrealmic 5th pentaspace pentrealm pentrealmic

tetrealm noun [George Olshevsky and Jonathan Bowers] - A four dimensional hyperplane; same as a flune.

tetrealmic adj [George Olshevsky and Jonathan Bowers] - Four-dimensional. Similar to tetral. See the chart under tetraspace.

tetronian noun [Aale de Winkel] - A person from tetraspace. adjective - From tetraspace. For example, a "tetronian river" is a river from tetraspace. See chart under tetraspace.

trength noun [Garrett Jones] - the four dimensional analog of length, width, and height. Something with great trength is trong, and something with small trength is tarrow. (These terms were previously tridth, tride, and trin.) Alternate terms are tetridth, tetride, and tetrin.

 dimension measure small measure great measure 1st length short long 2nd width narrow wide 3rd height short tall 4th trength tarrow trong

trionian noun [Aale de Winkel] - A person from realmspace. adjective - From realmspace. For example, a "trionian river" is a river from realmspace. See chart under tetraspace.

trong adj [Garrett Jones] - the four dimensional analog of long, wide, and tall. It is a measure of great trength. Its opposite is tarrow.

universe noun - All space, matter, and energy that occur within a single hyperplane; the space in which a particular object exists.

up adverb - The direction opposite of gravity; movement in the positive y direction. In realmspace, represented by the vector <0,1,0>. The opposite direction is down.  See the chart under direction.

upsilon adverb - One of the two directions pointing out of realmspace into tetraspace; movement in the positive w direction. In tetraspace, represented by the vector <0,0,0,1>. This term refers to the direction analogous to up; the 'u' in 'upsilon' corresponds to the 'u' in 'up'. In some literature, this direction is referred to as kata. This term is the same direction as wint, but is used in a different context: upsilon is movement from realmspace to tetraspace, but wint is a turning direction. The opposite direction is delta. See the chart under direction.

vertex noun - A zero-dimensional unit of a one-or-higher dimensional figure. A vertex is the intersection point of two edges. It is the zero-dimensional analog of edge, face, cell, and teron

volume noun - The measure of space within a three dimensional object. It is the realmspace analog of area in planespace and bulk in tetraspace.

west noun - The cardinal direction away from planetary rotation. The opposite direction is east. Distance east or west is measured in longitude

wint adverb [Jonathan Bowers] - One of the two extra turning directions of a tetronian object. In tetraspace, represented by the vector <0,0,0,1>. In realmspace there are two turning directions, left and right, but in tetraspace, there are two more. The four tetronian turning directions are located in a particular order depending on which direction you look at them. If you look at an object containing these directions and you look from the back of the object towards the front, then when you traverse them in a clockwise direction starting from the left, the order is left, wint, right, zant. Thus, wint is 90 degrees clockwise from left and 90 degrees counter-clockwise from right. Zant is the opposite: 90 degrees counter-clockwise from left and 90 degrees clockwise from right. See the chart under direction.

zakos noun [Oren] - The portion of an object that is furthest kata. It is the analog of the bottom portion of an object for the direction down, or the back portion for the backward direction. Derived from "zenith kata". The opposite side of an object is its apos.

zant adverb [Jonathan Bowers] - Opposite of wint. In tetraspace, represented by the vector <0,0,0,-1>. See wint for the description of the turning directions in tetraspace. See the chart under direction.

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