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-spherecrosses |
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**.