# DEFINITIONS

Agonal is a row, column, pillar, etc. of a magic figure.

Aspects refer to visually different arrangements of the numbers in a magic figure accomplished by changing the dimensional orientation (positions of the axes) or reflections (inversions). For an n-dimensional figure, there are n! possible dimensional orientations and 2n possible reflection combinations. Thus an n-dimensional figure has 2n x n! aspects.

Associated squares have all their complementary pairs located at the vectors (ai, bj) and (-ai, -bj) relative to the center of the square. For cubes, they are located at the vectors (ai, bj, ck) and (-ai, -bj, -ck) relative to the center of the cube, etc. These pairs are often described as being diametrically equidistant from the center of the figure. If the figure is an odd order then the central number is the midpoint of the numbers in the figure.

A base line for a magic figure of order-2p is composed of 2(p-q) identical base line units where p ≥ q. (See base line unit definition.) A base line for a magic figure of order-op is composed of o(p-q) identical base line units where p ≥ q.

Base line family is one of two groupings of the cubes C base lines. The members of a family are all compatible with each other but not with members of the other family. The families are C0, C3, C5, C6 and C1, C2, C4, C7. The families are often important in determining compatibility of base cubes.

Base line type is another grouping of the cubes C base lines. The C0 base line and all seven of its translations comprise one type and the C2 base line and all of its translations comprise the other. The base line types are useful for describing base cubes that cannot be made using base lines.

For figures made using binary math, a base line unit is a bit pattern of length 2q where q>=1 and bits spaced 2q/2 apart are inverses where inversion means 0 = 1 and 1 = 0. For figures made using odd prime bases, o, a base line unit is of length oq. It is composed of o parts of length o(q-1) and contains equal amounts of each number 0 to o-1. The first digit of each of the o parts of the base line unit must be different integers. This is also true for the second digit, the third, etc.

Base square, base cube, base tesseract, etc. refers to an order-2p square, cube, etc. consisting of an equal number of just zeros and ones. More generally for an order-op square, cube, etc. the base figure consists of an equal number of 0's, 1's, ... , (o-1)'s. To be a valid base square, etc. it must add to a magic constant in all ways that the target magic figure is expected to add to its magic constant. When the term is used on these pages, it always refers to a valid base figure.

Catchup magic squares have at least two consecutive base squares in which a feature is not predictable. The feature occurs in the magic square because imbalances in the base squares are canceled out so that the sum in the final magic square is the same as that obtained by combining predictable base squares.

Compact magic squares have every 2x2 square within the larger square including wrap around equal to S*4/m where S is the magic constant and m is the order of the square. A compact magic cube has every 2x2x2 sub-cube within the larger cube add to S*8/m, a compact magic tesseract has every 2x2x2x2 sub-tesseract add to S*16/m, etc. The more general definition of Aale de Winkel, {2compact3} will often be used. In this representation, the superscript indicates the number of dimensions in the figure and the subscript the spacing of numbers. The definition is further modified for ternary, quinery, etc. figures so that {2compact7}9 would indicate 9 numbers are summed. These numbers would be evenly spaced in a 7x7 square grid for the above example.

Compatible base lines are binary base lines that can be combined in one dimension such the result after adding 2 times one base line plus the second base line is a new base line with uniform integral distribution. The base line multiplied by 2 may already be a combination of base lines. The concept can be extended to ternary, etc. base lines.

Compatible base squares, cubes, tesseracts, etc. are binary base figures that can be combined such that the result after adding 2 times one base figure plus the second base figure is a new base figure with uniform integral distribution. The base figure multiplied by 2 may already be a combination of base figures. The concept can be extended to ternary, etc. base figures.

Compatible groups are a set of n compatible base lines where n is the number of dimensions of the figure. The base line resulting from 2n times the first base line plus 2n+1 times the second base line plus ... plus 2 times the n-1 base line plus the final base line must have uniform integral distribution.

Complement and Inverse can generally be used interchangeably on this site. In much of the magic square literature complementary numbers are defined as the pair of numbers that add to 2*S/m where S is the magic constant of the figure and m is its order. On most of this site, inverse numbers are the two numbers that when written in binary have all their bits different, i.e. 11 and 4 are inverses because the bits of their binary equivalents 1011 and 0100 are all different. For the magic figures made from binary base lines discussed on this site, the two terms are equivalent. This will not be true for all magic figures.

Complete for an even ordered magic square means that all of the complementary numbers in the square are located a (m/2, m/2) vector away where m is the order of the figure. For a cube they are located a (m/2, m/2, m/2) vector away, etc. This definition is modified for figures made with non-binary base lines. Thus for a figure made using ternary base lines there would be three numbers spaced evenly along the diagonal that always add to a constant and this will be designated as {complete3}. For a figure made using quinary base lines, there would be five evenly spaced numbers to combine or {complete5}, etc.

Dimensional orientation A magic square has two dimensions represented by the x and y axes. If the x- and y-axes and all the accompanying numbers are switched then a visually different magic square is seen. This visually different square is not a new magic square. It is the same square in a different dimensional orientation. A magic cube has three dimensions represented by the x-, y-, and z-axes. These axes can be rearranged into six different dimensional orientations. In general, an n-dimensional figure can be rearranged into n! different dimensional orientations.

Even integral distribution indicates that every integer in a magic figure occurs the same number of times. For a magic figure, there should be one of every integer. For binary base figures, there must be equal numbers of zeros and ones. For intermediate binary figures created during a build mode there will be the same multiple of two of each integer present. The concept can be extended to ternary, etc. figures.

Inverse confusingly is used to describe two different phenomena. See 'Complement and Inverse' for one definition. Inversion is also used to describe the process of translating every number in a cube through a point, line or plane to a new point the same distance but opposite side of the point, line or plane. The term is used to describe the visual process that accompanies this change when viewed using the cube generator in the 3-D mode. The change is more commonly called a reflection in magic cube literature but a reflection does not describe the visual seen with the generator.

Magic constant is the value that every row, column, etc. of the magic figure should add to. It is usually written symbolically as S.

Magic cube or magic tesseract on these pages will generally refer to nasik magic cubes or nasik magic tesseracts and not the myriad of other possible magic cubes or tesseracts.

Master base lines are the numbers in the lines of the magic figure that contain the zero and are parallel to one of the axes. Combining corresponding digits of the binary equivalents of sets of master base line numbers using an exclusive OR function will give the number in binary that occurs at the intersection of the set of master base line numbers that were combined. There are also ternary, quinary, etc. master base lines.

Nasik refers to magic figures in which all pan-n-agonals sum to the magic constant. This is a modern definition for a term that once had a somewhat less restrictive meaning.

Order when referring to a magic figure is usually written followed by a number such as order-4. The number indicates the size of the figure in every dimension.

Orthodox when referring to a magic figure or a feature of the magic figure, it means that the group is derived from base figures in which that group is always predictable.

Perfect when referring to a magic figure does not have a consistent definition in the literature. When used on these pages it refers to figures in which all pan-n-agonals sum to the magic constant. Much of the literature refers to perfect magic cubes that do not meet this requirement. The term nasik is preferred.

Predictable when referring to a base figure or a feature in the base figure, it means that the group has uniform integral distribution. (see also orthodox) If all the base figures that make up a magic figure have the same feature and the feature is predictable in all the base squares then the magic square made from those base square will have the feature as well.

Translation in a magic figure means that all numbers within the figure are moved to new locations within the figure using the same vector. Magic features are lost for most magic figures when the numbers are translated. Most figures discussed on these pages can be translated without losing major features. Features that cannot be translated throughout the figure can be lost. Use of the prefix pan indicates that the numbers can be translated any vector within the figure retaining the integrity of all agonals.

Uneven integral distribution indicates that the count of each individual integer in a magic figure is not the same for every integer in the figure. For a magic figure, there should be one of every integer in the range. If an integer occurs twice or not at all then there is uneven distribution. For intermediate figures created during a build mode there will be uneven integral distribution when not all of the integers occur the same number of times.

Uniform integral distribution indicates that the count of each individual integer in a magic figure or intermediate magic figure is the same for every integer in the figure. For a magic figure, there should be one of every integer in its range. For an intermediate magic figure created during construction every integer within the figure should be repeated the same number of times. For binary constructions, this number will be a multiple of 2. For ternary constructions it will be a multiple of 3, etc.

Unique cube, tesseract, etc. is a way to describe a group of cubes that share the same set of base cubes. Each order-8 magic cube is composed of nine base cubes. The nine multipliers of the base cubes can be placed in any order creating 9! different cubes. Each of these distinct cubes also have 48 aspects. All can be grouped using one unique code.

Wrap around is the ability to move one side (line for square, face for cube, etc.) of a figure to the directly opposite side without losing any of the magic features. It is an alternate way to describe a translation.