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.
Base line for a figure of order-4n is a series of 4n digits. A binary base line consists of equal numbers of zeros and ones in a pattern such that either the first and second halves are identical or they are inverses. Base lines are usually defined to start with a zero. Binary base lines are often shown using an alphanumeric code such as C3 where the letter indicates the length of a base line subunit. An A subunit is 2 bits long, a B subunit 4 bits, a C 8 bits, etc. The subscript becomes the binary code in the first half of the subunit and its inverse is the second half. If necessary the subunit is repeated until the base line is filled. Ternary base lines consist of equal numbers of 0's, 1's, and 2's. Quinary base lines equal numbers of 0's to 4's, etc.
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.
Base square, base cube, base tesseract, etc. refers to an order-2n square, cube, etc. consisting of an equal number of just zeros and ones. More generally for an order-mn square, cube, etc. the base figure consists of an equal number of 0's, 1's, ... , (m-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.
Compact magic squares have every 2x2 square within the larger square including wrap around equal to S*4/n where S is the magic constant and n is the order of the square. A compact magic cube has every 2x2x2 sub-cube within the larger cube add to S*8/n, a compact magic tesseract has exery 2x2x2x2 sub-tesseract add to S*16/n, etc.
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.
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 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.
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.
Magic cube or magic tesseract on these pages will generally refer to order-8 pan-2,3-agonal magic cube or order-16 pan-2,3,4-agonal magic tesseract and not the myriad of other possible magic cubes or tesseracts. The Upsized Figures section is an exception to this statement.
Magic square or pan-magic square when used without prefix normally infers the order-4 pan-magic squares as other squares are not often mentioned except in the Upsized Figures section.
Master base lines or magic lines are the numbers in the lines of the magic figure that contain the zero and are parallel to one of the axes. Combining 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 and quinary master base lines.
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.
Translation in a magic figure means that all number 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. The figures discussed on these pages can all be translated without losing major features. Some features described in the Upsized Figures section 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.