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A
Nasik (perfect) magic cube is pantriagonal and all
of its planes (the magic
On this page, comments referring
to magic cubes should be considered, |
This is my original page on Perfect Magic Cubes I have since posted another page that discusses this subject from a different point of view.
Introduction |
Why I feel this page is necessary and what is included to show the need for a revised definition. |
History |
Mr. Hendricks recounts the events in the development of the modern definition of perfect for magic objects. |
Definitions |
Definitions relevant to Perfect magic squares, excerpted from Magic Square Lexicon: Illustrated. |
Inter-dimensional Comparisons |
Tables compare features between hypercubes of different dimensions. |
Examples - Cubes |
The 4 'basic' order-3, a 'triagonal' order-4, an order-8 'perfect' and an order-9 'perfect' magic cube. |
Dimensions 4 and 5 |
A 'basic' order 3 and a 'quadragonal' order-4 magic tesseract, and information on two 'perfect' magic tesseracts. |
Conclusion and References |
A summary of this page, some relevant references, and some relevant links. |
In discussions regarding magic cubes, there seems to be much confusion about the term ‘perfect’.
Pandiagonal magic squares have long been referred to
as being perfect, presumably because they had additional lines (of m numbers) that summed
correctly.
Over the last 125 years or so, as work with magic cubes progressed and more features were
discovered, they have often been referred to as ‘perfect’.
Over the last 25 years or so, John Hendricks and
others have been doing more and more work with magic objects of 4 (tesseracts), 5, and
higher dimensions. When features between these different dimensions were compared by Mr.
Hendricks, it quickly became evident that the term ‘perfect’
should be redefined.
This he preceded to do, and since his retirement has been actively investigating and
refining his ideas.
In an effort to publicize his work and the new definitions, he has self published a number
of books.
I received an email from Mr. Hendricks on February 27, 2002 on the history of this definition. This was after I had mentioned to him about the large number of inquiries on the subject I was receiving through my Web pages.
The result is this page about perfect magic
hypercubes, with emphasis on hypercubes of dimension 3. I have included the modern definition for these cubes and other associated
definitions. Also shown are examples of orders 8 and 9 perfect cubes, and tables
that compare features common to hypercubes of the different dimensions.
The section closes with a short discussion of order 16 and order 32 perfect magic
tesseracts. These will further reinforce the relationship between ‘perfect’
hypercubes of different dimensions.
Addendum: After doing a survey of published magic cubes, I have published another page on perfect cubes. It serves as a supplement to this page, which I have left unchanged.
Addendum2 (Jan. 2005) I have added several definitions. Required because of the discovery of a sixth class of magic cube.
Addendum3 (May 2006) Expanded material on the term Nasik and emphasizing monagonal (or 1-agonal) as the preferred term to replace i-row.
This page uses m for order of the magic object and n for the dimension being referred to.
From an email attachment of March 2, 2002 from John Hendricks
General History.
Sixty years ago when we were kids you were lucky to have a magic cube in the first place. There were a few around, but no 7x7x7 cube existed although they had orders 3 to 10 except it. Andrews & Co. (1) set out the definition of a magic cube to sum in rows, columns, pillars and only the four space diagonals. Then they speculated on what would be considered a perfect cube. They all agreed that the continuous diagonals of a plane face, as an extra measure, would do as a bare minimum requirement. And that is about where I came into the picture. I eventually filled in the 7th order garden-variety cube and had it published. Nobody to my knowledge ever dreamt that it would be possible to get magic squares throughout a cube until Myers did so. (Editors note: Myers cube contained only simple magic squares.) So the definition changed for perfect cube to row, columns, pillars, the four triagonals and all the diagonals in all the planes including the broken ones. This allowed Collison and others to show an order 7 such cube. These are considered nowadays as pandiagonal magic cubes.
Meanwhile, working in isolation, apart from the mathematical community, I felt that such a cube is not equivalent to a pandiagonal magic square and went searching for the pantriagonal cube. I did not care about the planar diagonals, but did care about the broken triagonals, hence I came up with an order 4 such cube and had it published. Then, the critics advised me that someone else had such a cube on file at a university in the east and that I was not the first to make such a cube. It turned out that I was the first to publish such a cube.
Then, I got busy with the meteorological service and marriage and had no more time for several years to look into such mathematical matters, During that interval, Capt. Benson (2) came along and made a perfect 8th order magic cube, wrote a book and so forever more one would conclude that the matter was settled and the definition should be immediately updated for perfect cubes for all time. They referred to this cube (which contains 30 pandiagonal magic squares) as pandiagonally perfect. They called the Myers cube (which contains 30 simple magic squares) as perfect. It is now referred to as a diagonal magic cube.
Back at the drawing board, I concluded that a perfect magic tesseract would have to have everything working too. I did publish the first magic tesseract in one diagram and all 57 more, I did publish the first panquadragonal magic tesseract of order 4. But these were not perfect.
Eventually, after “retirement” I found the key and self-published the perfect magic tesseract of order 16 and the 5-dimensional magic hypercube of order 32 also in the same booklet (3), I also discounted the Cameron Cube (4), which was the name given to a special magic tesseract, because a simple examination of the triagonals of some of the cubes in the tesseract simply do not add up. I also published a perfect 8th order magic cube,” along with 9th and 11th order ones (5).
John
(1) Andrews, W.S., Magic Squares
& Cubes, Dover Publ., 1960 (original
publication Open Court,1917)
(2) Benson, W. & Jacoby, O., Magic
Cubes: New Recreations, , Dover Publ.,
1981, 0-486-24140-8
To help our understanding of the term ‘perfect’ as it applies to magic objects, I have included some appropriate definitions. These are taken from Magic Square Lexicon: Illustrated, but have been edited for brevity and most illustrations removed.
Compact Plus
When the eight corners of all orders of sub-cubes contained within a cube
including wrap-around, sum to 8S/m where m is the order of the
parent cube. I have adapted this term
from Gakuho Abe’s [1] term ‘compact’ which he used to indicate that all 2x2
squares of order-4 sum to S.
Kanji Setsuda [2] uses the term ‘composite’ for this feature in magic cubes but
I feel that this can cause confusion with ‘composite’ magic squares.
Complete
Every pantriagonal contains m/2 complement pairs, spaced m/2
apart. Note that this is a requirement for most-perfect magic squares. Coined by
Kanji Setsuda [2].
[1] Gakuho Abe, Fifty Problems of Magic
Squares, Self published 1950. Later republished in Discrete Math, 127, 1994,
pp 3-13.
[2] Kanji Setsuda’s Compact (composite) and Complete magic Cubes Web pages may
be accessed from here.
http://homepage2.nifty.com/KanjiSetsuda/pages/EnglishP1.html
Diagonal magic cube
A magic cube that has the additional feature that both main
diagonals of all 3m planar squares sum to S. Because all rows and columns
sum correctly as the original requirement for a magic cube, this means that all
3m orthogonal planes are simple magic squares. Some, but not all of these planar
squares may be pandiagonal magic. The six oblique squares will automatically be be
magic too. Order 5 is the smallest possible diagonal magic cube.
For a short time, I used the term 'myers' for this type of magic cube.
However, this name, I believe, is more suitable, and so will be used from now
on. The name diagonal magic cube was suggested by Aale de Winkel.
This class of cube was missed in Hendricks original unified classification.
The Myer's diagonal cube was popularized as 'perfect' by
Martin Gardner in Jan. 1976.
ALL singly-even orders of diagonal magic cubes are proper
(see definition for proper).
Christian Boyer and Walter Trump refer to this class as perfect. Trump has since started using the term strictly-magic to avoid confusion with Hendricks perfect.
i-row
An i-row is a row, column, pillar, file, etc.,
of an n-dimensional hypercube of order-m. Some authors refer to these as “the
orthogonals” because they are all mutually perpendicular to each other. (An i-row is
parallel to an xi-axis where the axes are numbered x1, x2,
x3, etc.)
A more modern term for i-row is
monagonal (or
1-agonal). It should be used instead of this term in future
references.
Magic
cube
An m x m x m array of cells with each cell containing a
number, usually an integer. These numbers are arranged so that the sum for each row, each
column, each pillar, and the four main triagonals are all the same. Note that it is not
required that the squares in the 3m planes of
the cube have correct diagonals.
These are the minimum requirements for a simple magic cube. All
magic cubes later defined have these features plus the additional
required features.
This was first (?) published in W. S. Andrews, Magic Squares
and Cubes, Open Court Publ., 1908 p 64.
This definition also appears in the better known edition 2 published in 1917,
also on page 64.
Magic
hypercube
An n-dimensional array of mn
cells containing the numbers1, 2, ¼, mn arranged in such a way that all rows,
columns, etc sum the magic sum, as well as the 2n-1
n-agonals. While usually used to refer to a higher dimension object, a square or cube may be considered
2 or 3-dimensional hypercube (respectively).
There are 2 classes or ranks of magic hypercubes of dimension two, 6 classes or ranks of magic hypercubes of dimension three, and 18 classes for dimension four, the tesseract?
Magic
Sum
The constant each row, column, etc., sums to is called the magic sum. It is denoted by S.
For a normal n-dimensional magic hypercube of
order-m the sum is m(mn + 1)/2.
Magic
tesseract
A magic tesseract is a four-dimensional array, equivalent to the magic
cube and magic square of lower dimensions, containing the numbers 1, 2, 3, …,
m4 arranged in such a way that the sum of the numbers
in each of the m3 rows, m3 columns, m3 pillars,
m3
files and in the eight major quadragonals passing through the center and joining opposite
corners is a constant sum S, called the magic sum, which is given by: S = m(m4+1)/2
and where n is called the order of the
tesseract.
Monagonal
A row, column, pillar, file, etc.,
of an n-dimensional hypercube of order-m. Some authors refer to these as “the
orthogonals” because they are all mutually perpendicular to each other. (A
monagonal (ie 1-agonal) is
parallel to an xi-axis where the axes are numbered x1, x2,
x3, etc.)
Customarily, a row runs from left to right; a column from front to back; a pillar runs up
and down and a file runs obliquely to the other three in the projection of a tesseract.
There are n(mn-1) monagonals in an n-dimensional hypercube of order-m.
If 1-agonal is used for rows , columns, etc., the nasik (perfect) magic
hypercube may then be defined as having all (including broken) r-agonals summing
correctly for r = 1 ... n.
Nasik cubes
In 1866, A. H. Frost introduced the term Nasik for magic squares
having the property that all monagonals and diagonals (including broken ones),
summed to the magic constant. These magic squares would later be called
pandiagonal or perfect.
In 1905, C. Planck extended the term to to refer to magic objects of any
dimension in which all r-agonals summed to the constant.
If all pan-r-agonals are correct for r = 1...n, we have a nasik (perfect)
magic hypercube of dimension n.
There is an extensive C. Planck quotation
here.
Nasik is unambiguous, and therefore the preferred term
for the often confusing perfect when referring to magic hypercube
classes.
(See perfect magic cube.)
n-agonals
n-agonal
(continuous): A line going from 1 corner, through the
center to the opposite corner, of a magic hypercube. For a cube or greater
dimension hypercube, this is sometimes called
a space diagonal.
There are 2n corners and 2n-1 n-agonals
in a magic hypercube.
n-agonal
(broken ): Lines parallel to a
continuous
n-agonal. For a two-dimensional object (a magic square) these lines will consist
of 2 segments totaling length m. For a cube, the line would consist of 2 or 3
segments, etc.
A 1-agonal (monagonal) is a line which crosses
only 1 dimension. A 2-agonal is
usually called a diagonal. It crosses two dimensions.
A 3-agonal is usually called a triagonal and crosses three dimensions, a quadragonal crosses
4 dimensions, etc. The variable r may be used instead of n to
indicate these other values for the agonal. (See r-agonal)
[1] First published mention was John R. Hendricks, The Pan-3-Agonal Magic Cube, JRM 5:1:1972, pp 51-54
|
For each continuous n-agonal in n-dimensional space, there are a number of broken n-agonals, depending upon the order of the hypercube. There are 2 continuous diagonals in a square, 4 continuous triagonals in a cube, and 8 continuous quadragonals in a tesseract. So, the numbers in the table must be multiplied by the number of continuous ones in order to determine how many and of which kind of n-agonals are in a hypercube. |
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Pandiagonal
Pandiagonal means “all diagonal”, which signifies that the broken diagonals are
also included. Sometimes pan-2-agonal is used, instead, especially in n-dimensional space.
A 2-agonal is described through space if any two coordinates change while the rest remain
constant.
For example in a cube of order 4, one could describe a diagonal through (1,2,3) by holding
y constant while x and z is allowed to change. Such a set could be:
(1,2,3) ; (4,2,4) ; (3,2,1) ; and (2,2,2)
In this example x is decreasing in increments of one and z is increasing by increments of
one and all coordinates are kept within the modulus 4. There are N = n!.mn-1/(n-2)! diagonals in an n-dimensional magic hypercube of
order m, including the broken ones.
If all pan-r-agonals are correct for r = 1 and 2, we have a pandiagonal magic
square, If r = 1, 2, and n, a pandiagonal hypercube of dimension n.
Pandiagonal
magic cube
A Pandiagonal Magic Cube has the normal requirements of a magic cube plus
the additional one that all the squares parallel to the sides of the cube
(planar squares) also be pandiagonal.
Remember that an ordinary magic cube does not require even the main diagonals of these
squares to be correct.
There are 9m2 + 4 lines that sum
correctly (m2 rows, m2 columns, m2 pillars, 4 main triagonals and 6m2 Diagonals).
It contains 3m pandiagonal
magic squares and 6 oblique squares, 0 to 3 of which are pandiagonal magic and
the others simple magic
Order-7 is the smallest possible order of pandiagonal magic cube. It contains 27
pandiagonal magic squares
(3 x 7 + 6).
M. Gardner referred to H. Langman's 1962 pandiagonal magic
cube as a perfect magic cube at the
same time he was calling the Myer's diagonal cube
perfect!
Presumably he did not recognize the difference between the three types of cubes.
Pandiagonal
magic square
Also known as Diabolic, Nasic, Continuous, Indian, Jaina or Perfect. To be pandiagonal, the broken diagonal pairs must
also sum to the constant. This is considered the top class of magic squares. There are 4m lines that sum correctly (m rows, m
columns and 2m diagonals).
Panquadragonal
Broken quadragonal pairs that are parallel to a quadragonal and that sum
to the magic constant. If all these pairs sum correctly, the magic tesseract is
panquadragonal. It is analogous to a pandiagonal magic square but instead of moving a row
or column from one side to the other and retaining the magic properties, you move any cube
from one side to the other. When one moves along the panquadragonal, 1 cell at a time,
four coordinates change. See also, Pantriagonal.
Pantriagonal
Sometimes called Pan-3-agonal.
This term is used for cubes, or high dimensional hypercubes. In n-dimensional space, if
any three coordinates are changing while the rest remain constant, then one describes a
triagonal through space, of which most are broken. The main triagonal is the one which
passes through (1,1,1) and has successive coordinates (2,2,2),…, (m,m,m) in a cube.
In N-dimensional space, the n-agonal may be broken into as many as n segments. For magic
cubes there are:
4 continuous triagonals
12(m-1) triagonals broken into pairs, and
4(m-2)(m-1) triagonals broken into 3 sections.
If all the broken Triagonal lines sum correctly, the magic cube is pantriagonal.
Pantriagonal
magic cube [1]
If all triagonal pairs and triplets (pan-triagonals) sum correctly, the magic cube is
pantriagonal. It is analogous to a pandiagonal magic square but instead of moving a row or
column from one side to the other and maintaining the magic properties, you may move any
plane from one side to the other.
There are 7m2 lines that sum
correctly (m2 rows, m2 columns, m2 pillars, and 4m2 triagonals). There may be some
diagonals in the cube, but they are not required. There may also be some magic
squares, either simple or pandiagonal, but they also are not required. Order-4 is the smallest possible order
pantriagonal magic cube. See also, Pandiagonal magic cube.
[1] First published mention was John R. Hendricks, The Pan-3-Agonal Magic Cube, JRM 5:1:1972, pp 51-54
Pantriagonal Diagonal – A magic cube that is a
combination Pantriagonal and Diagonal cube. All main and broken triagonals
must sum correctly, In addition, it will contain 3m order m
simple magic squares in the orthogonal planes, and 6 order m
pandiagonal magic squares in the oblique planes.
For short, I will reduce this unwieldy name to PantriagDiag. This is number
4 in what is now 6 classes of magic cubes. So far, very little is known of
this class of cube. The only ones constructed so far (both by Nakamura) are
order 8 (not associated and associated).
This cube was discovered by Mitsutoshi Nakamura and named by him in 2004
Pathfinder
An orderly and systematic way to find one’s way through n-dimensional
space. Through any given element, or cell, there are
(3n-1)/2 different
paths., or lines. For a square, this means that there are 4 paths, which are a row, a
column and two (broken, if needed) diagonal ways. Through any cell of a cube, there are 13 routes. Through a tesseract, there are 40. One
may travel forwards, or backwards on any route, or path. The method is found in Magic Squares to Tesseracts by Computer. Hendricks uses this method to show the numbers
contained in his higher order magic hypercubes (via simple computer programs).
Perfect
magic cube
A perfect magic cube is pantriagonal and all of its planes (the magic
squares) are pandiagonal. In a perfect magic cube there are 9m
pandiagonal magic squares. That is, all 3m orthogonal planes, the 6
oblique planes, and the 6(m-1) broken planes parallel to the oblique
planes. There are 13 m2
lines that sum correctly (m2 rows, m2 columns, m2 pillars, 4m2 triagonals and 6m2 diagonals).
Order-8 is the smallest
possible order perfect magic cube. Perfect magic cubes of orders 10, 14, 18,
etc. and 12, 20, etc. are impossible. this was proved by Rosser and Walker in
1939, and by Stertenbrink and de Winkel in 2004. . See
Pandiagonal
Impossibility Proof.
*** This is a new definition. ***
Examples of an older definition of a Perfect Magic Cube was the Frankenstein 1875, Myers 1970
order 8 cubes.
These cubes contained 3m simple magic squares and are now referred
to as Diagonal magic cubes. Perfect is now construed to mean
that the cube is pandiagonal and pantriagonal, and all lower order magic objects
(i.e. squares) within it are perfect. This makes the definition consistent for all dimensions.
See the definition (above) for pandiagonal and pantriagonal magic cubes .
This is also consistent with C. Planck's (1905) revised definition for A. H. Frost's
(1866) term Nasik . (See Perfect magic hypercube)
Mitsutoshi Nakamura uses the term pan-2,3-agonal and nasik instead of perfect to avoid confusion. He also uses the term strictly-magic for the diagonal class, to avoid confusion over Boyer's perfect.
Perfect magic
hypercube
A hypercube of dimension n is perfect if all pan-n-agonals sum
correctly, and all lower dimension hypercubes contained in it are perfect.
Through every cell on the perfect hypercube there are (3n-1)/2
different routes that must sum the magic sum.
As per Dr. C. Planck (1905), these hypercubes are of type Nasik. The
pandiagonal magic square is a perfect hypercube of dimension 2 (and was called
Nasik by Dr. A.H. Frost in 1866).
If all pan-r-agonals are correct for r = 1...n, we have a nasik (Hendricks
perfect)
magic hypercube of dimension n.
Because nasik in unambiguous, it
is preferred instead of the often confusing perfect.
Perfect
magic square
Another traditional (but now not commonly used) name for Pandiagonal magic square.
However, this name shows the relationship of the highest class of rectilinear magic
figures, the perfect square, perfect cube, perfect tesseract, etc. See the comment re nasik in
perfect magic hypercube.
Perfect
magic tesseract
A tesseract is a 4-dimensional hypercube. It is perfect if all
pan-quadragonals are correct, and all the magic squares and magic cubes within it are
perfect. i.e. the magic squares are all pandiagonal and the magic cubes are all
pantriagonal and pandiagonal. There are 40m2
lines that sum correctly. They are m3
rows, m3 columns, m3 pillars, m3 files, 8m3 quadragonals, 16m3 triagonals, and 12m3 diagonals.
John R. Hendricks constructed the first perfect magic tesseract (order-16) in 1998. It was
confirmed correct by Clifford Pickover in 1999. He later published the equations for a
5-dimensional perfect magic hypercube of order-32. However, as it contains the numbers 1
to 33,554,432, he thought it impractical to publish the hypercube itself!! The numbers in
these two hypercubes may be inspected using his simple Pathfinder programs.
If all pan-r-agonals are correct for r = 1...n, we have a nasik (perfect)
magic hypercube of dimension n.
Because nasik in unambiguous, it
is preferred instead of the often confusing perfect.
Proper
When applied to a magic cube of a particular class, means that this cube
contains only the minimum features required for that class.
For example: a simple magic cube requires that no planes within it be magic
squares. However, some planes may be magic and the cube is still called simple,
because the next classification requires that ALL planes be simple magic
squares.
So if a cube contains NO magic squares, it may then be referred to as a
proper simple magic cube!
Likewise a proper pantriagonal magic cube would contain NO magic squares,
a proper diagonal magic cube would contain NO pandiagonal magic squares,
etc.
This term was suggested by Mitsutoshi Nakamura in an Apr. 15, 2004 email.
He also reported that all singly-even orders of diagonal magic cubes are proper!
r-agonal
Term used to indicate a group of agonals (Diagonal, triagonal, etc).
(See n-agonal)
Examples:
If all r-agonals are correct for r = 1 and 2, we have the minimum qualifications
for a simple magic square.
If all pan-r-agonals are correct for r = 1...n, we have a nasik (Hendricks perfect)
magic hypercube of dimension n.
Simple magic hypercube
A magic square, cube, tesseract, etc., where all orthogonal lines plus all
space diagonals sum correctly.
If all r-agonals are correct for r = 1 and 2, we have the minimum
qualifications for a simple magic square. If r = 1 and n, a simple magic
hypercube of n dimensions!
Space diagonal
A line that goes from a corner of a magic hypercube, through the center, to
the opposite corner. See (n-agonal)
Triagonal
A space diagonal that goes from 1 corner of a magic cube to the opposite
corner, passing through the center of the cube. There are 4 of these in a magic cube and
all must sum correctly (as well as the rows, columns and pillars) for the cube to be
magic. As you go from cell to cell along the line, all three coordinates change. In
tesseracts this is called a quadragonal. For higher order hypercubes, this is called an n-agonal or space diagonal. Of course, with these
higher dimensions there are more coordinates. A triagonal is sometimes called a long
diagonal. See n-agonal
Summations
The magic sum for an n-Dimensional Magic Hypercube of Order m is given by: S = m(1 + mn)/2
In a magic object, there are many lines that produce the magic sum.
The table below, shows the minimum requirement of the number of lines for various types of
magic hypercubes and is derived from the following equation:
N = 2(r-1)n!m(n-1)/[r!(n-r)!]
Where:
N is the number of r-agonals
n is the dimension of the hypercube
m is the order of the hypercube, and
r is the dimension of the hyperplane.
When r = 1, the number
of orthogonals is given by N. As well, shown is the smallest order for the various
classifications of pandiagonal, pantriagonal, etc. which is known. for each dimension. Some of the tesseracts
are not known yet and some of these varieties have not been constructed yet.
This table provides the minimum requirements for each category. Usually, there are some
extra lines, which may sum the magic sum, but not a complete set so as to change the
category.
| n-agonals | Lines | |||||||
| Magic Hypercube | n | Lowest Order |
i - rows |
2 | 3 | 4 | Total | Equivelent order 8 |
| Square | 2 | |||||||
| Simple | 3 | 2m | 2 | 2m + 2 | 18 | |||
| Perfect | 4 | 2m | 2m | 4m | 32 | |||
| Cube | 3 | |||||||
| Simple | 3 | 3m2 | 4 | 3m2 + 4 | 196 | |||
| Diagonal | 5 | 3m2 | 6m | 4 | 3m2+6m+ 4 | 244 | ||
| Pantriagonal | 4 | 3m2 | 4m2 | 7m2 | 448 | |||
| PantriagDiag | 8? | 3m2 | 6m | 4m2 | 3m2+6m+ 4m2 | 452 | ||
| Pandiagonal | 7 | 3m2 | 6m2 | 4 | 9m2 + 4 | 580 | ||
| Perfect | 8 | 3m2 | 6m2 | 4m2 | 13m2 | 832 | ||
| Tesseract | 4 | |||||||
| Simple | 3 | 4m3 | 8 | 4m3 + 8 | 2056 | |||
| Panquadragonal | 4 | 4m3 | 8m3 | 12m3 | 6144 | |||
| Pandiagonal | ? | 4m3 | 12m3 | 8 | 16m3 + 8 | 8200 | ||
| Pantriagonal | ? | 4m3 | 16m3 | 8 | 20m3 + 8 | 10240 | ||
| Pan2 + pan4 | ? | 4m3 | 12m3 | 8m3 | 24m3 | 12296 | ||
| Pan3 + pan4 | ? | 4m3 | 16m3 | 8m3 | 28m3 | 14336 | ||
| Pan2 + pan 3 | ? | 4m3 | 12m3 | 16m3 | 8 | 32m3 + 8 | 16392 | |
| Perfect | 16 | 4m3 | 12m3 | 16m3 | 8m3 | 40m3 | 20480 | |
Hypercubes – number of correct summations. (1) |
||||||||
(1) H. D. Heinz and J. R. Hendricks, Magic Square Lexicon: Illustrated, 2000, 0-9687985-0-0, page 165
| Comparison - Correct Summations Required | ||
| magic Square Regular |
Magic Cube Regular |
Magic Tesseract Regular |
| m rows | m2 rows | m3 rows |
| m columns | m2 columns | m3 columns |
| 2 diagonals | m2 pillars | m3 pillars |
| 4 3-agonals | m3 files | |
| 8 4-agonals | ||
| Perfect | Perfect | Perfect |
| m rows | m2 rows | m3 rows |
| m columns | m2 columns | m3 columns |
| 2m diagonals | m2 pillars | m3 pillars |
| 4m2 3-agonals | m3 files | |
| 6m2 2-agonals | 8m3 4-agonals | |
| 12m3 3-agonals | ||
| 16m3 2-agonals | ||
| For a normal n-dimensional magic hypercube of
order-m, the sum is m(mn+1)/2 |
||
| Magic Squares, Cubes and Tesseracts Compared (2) | ||
(2) H. D. Heinz and J. R. Hendricks, Magic Square Lexicon: Illustrated, 2000, 0-9687985-0-0, page 90
| Hyperplanes contained in a hypercube | |||||
| Dimension | i-rows | Squares | Cubes | Tesseracts | 5-D Hyper. |
| 2 | 2m | 1 | 0 | 0 | 0 |
| 3 | 3m2 | 3m | 1 | 0 | 0 |
| 4 | 4m3 | 6m2 | 4m | 1 | 0 |
| 5 | 5m4 | 10m3 | 10m2 | 5m | 1 |
| 6 | 6m5 | 15m4 | 20m3 | 15m2 | 6m |
| 7 | 7m6 | 21m5 | 35m4 | 35m3 | 21m2 |
An n-dimensional array of mn cells containing the numbers1, 2,
..., mn arranged in such a way that
all rows, columns, etc sum the magic sum, as well as the 2n-1 n-agonals.
Remember that i-rows are orthogonals only. Correct n-agonals are not shown in this table.
If the hypercube is perfect, all these hyper-planes must also have all the
n-agonals summing correctly.
(3) H. D. Heinz and J. R. Hendricks, Magic Square Lexicon: Illustrated, 2000, 0-9687985-0-0, page 80
Hypercubes cut by n hyperspaces |
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| A magic square may be cut by a 1-dimensional magic i-row in 2 ways |
A magic cube may be cut by a 2 -dimensional magic square in 3 ways. |
A magic tesseract may be cut by a 3 -dimensional magic cube in 4 ways. |
If you are wondering where the
cube in the fourth direction (in the tesseract) is, it is formed by the
center square planes in the 3 horizontal planes of cubes. |
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The 4 Basic Order-3 Magic Cubes |
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| Catalogue numbers 1-15-17-23 2-15-18-24 4-17-18-26 and 6-16-17-26 Each of these four may be viewed in 48 aspects due to rotations and reflections. There is 1 basic order-3 magic square with 8 aspects and 58 basic magic tesseracts, each with 384 aspects due to rotations and reflections. The 3 squares that bisect each of these four cubes are also magic although that is not a requirement of a simple magic cube.. |
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Pantriagonal magic Cube |
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05 58 08 59
28 39 25 38 53 10 56 11
44 23 41 22 This is text format. Best used for larger magic cubes. |
Pantriagonal is the
lowest class of magic cube (next to the simple and the associated). Lines are required to
sum correctly in 7m2 ways: m2 rows, m2
columns, Just as a complete row or column may be moved from 1 side to the other of a pandiagonal magic square, so too can a square plane be moved to the opposite side of this pantriagonal magic cube without destroying the triagonals. J. R. Hendricks, Magic Squares to Tesseracts by Computer, Self-published 1999, 0-9684700-0-9, page 70 |
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A perfect magic cube must
add up in all possible ways. There are 13 such ways through each element.
Looking at it the other way, there are a total of l3m2 paths that sum correctly.
m2 rows
m2 columns
m2 pillars
4m2 triagonals
6m2 diagonals
Order-8 is the smallest possible magic cube that can be perfect, with 832 lines summing to 2052. This one by J. R. Hendricks is an example of such a cube.
Horizontal plane I - Top II 88 185 240 449 408 377 304 1 395 350 307 38 75 158 243 486 300 5 84 189 236 453 404 381 242 487 394 351 306 39 74 159 403 382 299 6 83 190 235 454 73 160 241 488 393 352 305 40 234 455 402 383 298 7 82 191 309 36 77 156 245 484 397 348 81 192 233 456 401 384 297 8 398 347 310 35 78 155 246 483 301 4 85 188 237 452 405 380 247 482 399 346 311 34 79 154 406 379 302 3 86 187 238 451 80 153 248 481 400 345 312 33 239 450 407 378 303 2 87 186 308 37 76 157 244 485 396 349 III IV 65 152 249 496 385 344 313 48 422 331 286 51 102 139 222 499 317 44 69 148 253 492 389 340 223 498 423 330 287 50 103 138 390 339 318 43 70 147 254 491 104 137 224 497 424 329 288 49 255 490 391 338 319 42 71 146 284 53 100 141 220 501 420 333 72 145 256 489 392 337 320 41 419 334 283 54 99 142 219 502 316 45 68 149 252 493 388 341 218 503 418 335 282 55 98 143 387 342 315 46 67 150 251 494 97 144 217 504 417 336 281 56 250 495 386 343 314 47 66 151 285 52 101 140 221 500 421 332 V VI 112 129 216 505 432 321 280 57 435 358 267 30 115 166 203 478 276 61 108 133 212 509 428 325 202 479 434 359 266 31 114 167 427 326 275 62 107 134 211 510 113 168 201 480 433 360 265 32 210 511 426 327 274 63 106 135 269 28 117 164 205 476 437 356 105 136 209 512 425 328 273 64 438 355 270 27 118 163 206 475 277 60 109 132 213 508 429 324 207 474 439 354 271 26 119 162 430 323 278 59 110 131 214 507 120 161 208 473 440 353 272 25 215 506 431 322 279 58 111 130 268 29 116 165 204 477 436 357 VII VIII - Bottom 121 176 193 472 441 368 257 24 414 371 294 11 94 179 230 459 261 20 125 172 197 468 445 364 231 458 415 370 295 10 95 178 446 363 262 19 126 171 198 467 96 177 232 457 416 369 296 9 199 466 447 362 263 18 127 170 292 13 92 181 228 461 412 373 128 169 200 465 448 361 264 17 411 374 291 14 91 182 227 462 260 21 124 173 196 469 444 365 226 463 410 375 290 15 90 183 443 366 259 22 123 174 195 470 89 184 225 464 409 376 289 16 194 471 442 367 258 23 122 175 293 12 93 180 229 460 413 372
J. R. Hendricks, Magic Squares to Tesseracts by Computer, Self-published 1998, 0-9684700-0-9, pp 77-80
Top II 52 442 588 339 495 233 704 158 274 494 227 707 157 277 46 444 591 342 104 256 7 469 624 402 522 215 686 627 405 521 209 689 103 259 1 471 170 713 140 319 34 451 606 348 504 28 453 609 351 503 164 716 139 322 384 567 197 695 122 265 16 406 633 121 268 10 408 636 387 566 191 698 433 615 366 513 179 650 149 301 79 173 653 148 304 73 435 618 369 512 283 25 415 570 393 549 242 677 131 396 548 236 680 130 286 19 417 573 659 86 310 61 478 597 375 531 188 480 600 378 530 182 662 85 313 55 558 224 722 113 292 43 424 579 330 295 37 426 582 333 557 218 725 112 642 357 540 206 668 95 247 70 460 671 94 250 64 462 645 360 539 200 III IV 271 48 447 594 341 488 230 706 160 335 491 229 709 154 273 51 450 593 688 106 253 3 474 630 404 515 212 477 629 398 518 211 691 100 255 6 497 167 715 142 316 30 456 612 350 318 33 459 611 344 500 166 718 136 639 386 560 194 697 124 262 12 411 700 118 264 15 414 638 380 563 193 75 438 621 368 506 176 652 151 298 509 175 655 145 300 78 441 620 362 133 280 21 420 576 395 542 239 679 575 389 545 238 682 127 282 24 423 185 661 88 307 57 483 603 377 524 60 486 602 371 527 184 664 82 309 332 551 221 724 115 289 39 429 585 109 291 42 432 584 326 554 220 727 465 648 359 533 203 670 97 244 66 202 673 91 246 69 468 647 353 536 V VI 156 276 54 449 587 338 490 232 703 590 337 493 226 705 159 279 53 443 214 685 102 258 9 476 623 401 517 8 470 626 400 520 208 687 105 261 347 499 169 712 138 321 36 458 605 141 324 35 452 608 346 502 163 714 413 632 383 562 196 694 120 267 18 190 696 123 270 17 407 635 382 565 303 81 440 614 365 508 178 649 147 364 511 172 651 150 306 80 434 617 676 129 285 27 422 569 392 544 241 416 572 391 547 235 678 132 288 26 526 187 658 84 312 63 485 596 374 315 62 479 599 373 529 181 660 87 578 329 553 223 721 111 294 45 431 723 114 297 44 425 581 328 556 217 72 467 641 356 535 205 667 93 249 538 199 669 96 252 71 461 644 355 VII VIII 708 162 278 47 446 589 340 487 228 445 592 334 489 231 711 161 272 50 514 210 690 108 260 2 473 625 403 254 5 472 628 397 516 213 693 107 607 349 496 165 717 144 323 29 455 720 143 317 32 454 610 343 498 168 11 410 634 385 559 192 699 126 269 561 195 702 125 263 14 409 637 379 153 305 74 437 616 367 505 174 654 619 361 507 177 657 152 299 77 436 237 681 135 287 20 419 571 394 541 23 418 574 388 543 240 684 134 281 376 523 183 663 90 314 56 482 598 89 308 59 481 601 370 525 186 666 428 580 331 550 219 726 117 296 38 222 729 116 290 41 427 583 325 552 251 65 464 643 358 532 201 672 99 352 534 204 675 98 245 68 463 646 IX 234 710 155 275 49 448 586 336 492 399 519 216 692 101 257 4 475 622 457 604 345 501 171 719 137 320 31 266 13 412 631 381 564 198 701 119 656 146 302 76 439 613 363 510 180 546 243 683 128 284 22 421 568 390 595 372 528 189 665 83 311 58 484 40 430 577 327 555 225 728 110 293 92 248 67 466 640 354 537 207 674
This cube is by John
Hendricks [1].
Order 9 is the lowest order normal cube that can be perfect and also associated
(although this cube is not associated).[2]
[1] J. R. Hendricks, Magic Squares to Tesseracts by Computer, Self-published 1999, 0-9684700-0-9, pp 81-84.
[2] W. S. Andrews, Magic Squares & Cubes, 2nd edition, Dover Publ. 1960, 9 (1917) Dr. Planck, p 366.
| Basic Order-3 Magic Tesseract MT#9 (Index # 54) | |
 |
A magic tesseract is a dimension-4 hypercube. To be simply magic, it
is required only that all rows, columns, pillars, files, and the 8 quadragonals sum
correctly. One of the authors (Hendricks) has found and all 58 basic magic tesseracts of order-3. He lists and displays illustrations of each of them in his book All Third-Order magic Tesseracts using the following indexing method: ·
Identify
the lowest of the 16 corner numbers. In this figure, the lowest corner number is 12 and the four numbers adjacent to it are 52, 61,62, and 76. Taking them in order; row, column, pillar and file they are already in ascending order, and, because the lowest corner is in the bottom left position we realize this tesseract is in the standard position. This definition is consistent with that of the Basic magic cube. This tesseract (and each of the 57 others) can be displayed in 383 other aspects (orientations). These are not considered unique solutions. John refers to this solution as MT#9 because it is the ninth order-3 tesseract he constructed. H. D. Heinz and J. R. Hendricks, Magic Square Lexicon: Illustrated, 2000, 0-9687985-0-0, page 14 |
A PAN-4-AGONAL MAGIC TESSERACT OF ORDER 4 The ability to translocate a row, or
column of a magic square from one side of the square to the other side without destroying
the diagonals is the hallmark of being pandiagonal. |
NOTE that the squares and cubes mentioned in the above quotation are themselves not magic. The diagonals of the squares sum incorrectly, as do the triagonals of the cubes. Only the rows, columns, pillars and files and the 8 quadragonals are required to sum correctly for a simple magic tesseract. This is exactly the equivalent of a simple magic cube which requires only that the rows, columns, pillars and 4 main triagonals.
The quadragonal magic
tesseract is the next higher class after the simple magic tesseract.
Required to sum correctly for this class of magic tesseract :
m3 rows
m3 columns
m3 pillars
m3 files
8m3 quadragonals
Equals
12 m3 ways that sum correctly (other ways may sum correctly but are not
required).
Text format for the above Order-4 Quadragonal Magic Tesseract |
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| 239 | 116 | 30 | 129 | 56 | 154 | 197 | 107 | 210 | 77 | 35 | 192 | 9 | 167 | 252 | 86 | |||
| 153 | 199 | 108 | 54 | 79 | 36 | 190 | 209 | 168 | 250 | 85 | 11 | 114 | 29 | 131 | 240 | |||
| 34 | 189 | 211 | 80 | 249 | 87 | 12 | 166 | 31 | 132 | 238 | 113 | 200 | 106 | 53 | 155 | |||
| 88 | 10 | 165 | 251 | 130 | 237 | 115 | 32 | 105 | 55 | 156 | 198 | 191 | 212 | 78 | 33 | |||
| 4, x, y, 1 | 4, x, y, 2 | 4, x, y, 3 | 4, x, y, 4 | |||||||||||||||
| 138 | 229 | 123 | 24 | 109 | 51 | 160 | 194 | 183 | 220 | 70 | 41 | 84 | 14 | 161 | 255 | |||
| 52 | 158 | 193 | 111 | 218 | 69 | 43 | 184 | 13 | 163 | 256 | 82 | 231 | 124 | 22 | 137 | |||
| 71 | 44 | 182 | 217 | 164 | 254 | 81 | 15 | 122 | 21 | 139 | 232 | 157 | 195 | 112 | 50 | |||
| 253 | 83 | 16 | 162 | 23 | 140 | 230 | 121 | 196 | 110 | 49 | 159 | 42 | 181 | 219 | 72 | |||
| 3, x, y, 1 | 3, x, y, 2 | 3, x, y, 3 | 3, x, y, 4 | |||||||||||||||
| 19 | 144 | 226 | 125 | 204 | 102 | 57 | 151 | 46 | 177 | 223 | 68 | 245 | 91 | 8 | 170 | |||
| 101 | 59 | 152 | 202 | 179 | 224 | 66 | 45 | 92 | 6 | 169 | 247 | 142 | 225 | 127 | 20 | |||
| 222 | 65 | 47 | 180 | 5 | 171 | 248 | 90 | 227 | 128 | 18 | 141 | 60 | 150 | 201 | 103 | |||
| 172 | 246 | 89 | 7 | 126 | 17 | 143 | 228 | 149 | 203 | 104 | 58 | 67 | 48 | 178 | 221 | |||
| 2, x, y, 1 | 2, x, y, 2 | 2, x, y, 3 | 2, x, y, 4 | |||||||||||||||
| 118 | 25 | 135 | 236 | 145 | 207 | 100 | 62 | 75 | 40 | 186 | 213 | 176 | 242 | 93 | 3 | |||
| 208 | 98 | 61 | 147 | 38 | 185 | 215 | 76 | 241 | 95 | 4 | 174 | 27 | 136 | 234 | 117 | |||
| 187 | 216 | 74 | 37 | 96 | 2 | 173 | 243 | 134 | 233 | 119 | 28 | 97 | 63 | 148 | 206 | |||
| 1 | 175 | 244 | 94 | 235 | 120 | 26 | 133 | 64 | 146 | 205 | 99 | 214 | 73 | 39 | 188 | |||
| 1, x, y, 1 | 1, x, y, 2 | 1, x, y, 3 | 1, x, y, 4 | |||||||||||||||
The colored cells indicate the corners of the tesseract,
 Corner values for the order 16 Perfect
magic tesseract. |
Required to
sum correctly for this highest class of dimension 4 magic hypercube: . The order-16 is
the smallest possible perfect magic tesseract. 49 152 diagonals |
| This perfect tesseract and the order 32 perfect dimension-5 hypercube he later constructed are obviously too large to publish here. The numbers in these two hypercubes may be inspected using his simple Pathfinder programs. | |
More information on this order 16 and an order 6 Inlaid magic tesseract may be seen on John R. Hendricks web site.
This magic hypercube was constructed by John Hendricks in 2000.
| There are: 33,554,432 numbers in it 5,242,880 i-rows (rows, columns, etc) 327,680 perfect magic squares 10,240 perfect magic cubes 160 perfect magic tesseracts (3) J.R.Hendricks, Perfect n-Dimensional Magic Hypercubes of Order 2n, Self-published,1999, 0-9684700-4-1, page 25. |
Which translates to: 5,242,880 i-rows (monagonal) 20,971,520 diagonals 41,943,040 3-agonals (triagonals) 41,943,040 4-agonals (quadragonals) 16,777,216 5-agonals for a total of 126,877,696 lines that sum to 536,870,928 |
Obviously, magic objects with this many numbers cannot be printed out for visualization or study. Mr. Hendricks has come up with the concept of pathfinders. These are relatively simple computer programs that generate the numbers that appear in any magic line through the hypercube. Conversely, the programs may be used to find the coordinate position of a given number in the magic hypercube.
Using the tesseract as an example: To show any row, column, or whatever you want. Pick any starting position (w,x,y,z) and any route from 1 to 40 and enter the values into the computer. Out come the numbers and the sum. If you want the reverse direction enter the route as a negative number (–5 instead of 5) and the numbers will all be reversed. For coordinates, 0 is the same as 16. For routes, zero is not a route.
Some of the routes are:
Route #1 is a pillar. Z-changes.
Route #3 is a column. Only y changes.
Route #9 is a row. Only x changes.
Route #27 is a file. Only w changes.
. . .
Route #40 is the main quadragonal.
On this page I have attempted to explain the meaning and reasoning
behind the term ‘perfect’ when applied to magic hypercubes.
In an attempt to avoid confusion, I have included definitions for related terms.
I have expanded on this idea with illustrations and tables, including 2 orders of perfect
magic cubes.
Much of the material for this page was obtained from the two books
below by John Hendricks.. I thank him for giving his permission to do so. These books,
however, go into much greater detail and should be referred to if a more in-depth
explanation is desired.
Over many years, Mr. Hendricks has contributed much to the study of higher order magic
hypercubes. However, He has also done a lot of original work with the lowly magic square,
especially in the development of unique variations. I encourage you to see more of his
work via the link below. Also much of his work is featured on
my J. R. Hendricks
page.
J. R. Hendricks, Perfect n-Dimensional Magic Hypercubes of
Order 2n, Self-published,1999, 0-9684700-4-1.
36 pages of theory,
equations and discussion on the definition of perfect as related to magic objects.
Also pathfinder basic programs for order-16 and 32 perfect
dimension 4 and 5 hypercubes
J. R. Hendricks, Magic Squares to Tesseracts by Computer, Self-published 1999,
0-9684700-0-9
212 pages of text, theory,
appendices, diagrams , etc, including basic program listings.
These books are now out of print (October, 2005). You can find more information
here about John Hendricks.
H. D. Heinz and J. R. Hendricks, Magic Square Lexicon:
Illustrated, 2000, 0-9687985-0-0
239 definitions, most with
illustrations, dealing with a large variety of magic objects.
More information and how to order the book is at
BookSale.htm
There are many books available on the subject of magic cubes. Here are four good ones.
Clifford A. Pickover, The Zen of Magic Squares, Circles, and
Stars, Princeton Univ. Pr., 2002, 0-691-07041-5
A great new book destined to become a classic on magic squares, cubes,
etc.
Specifically mentioned on page 101 is the new requirements for a cube
to be considered 'perfect
W. H. Benson and O. Jacoby, Magic Cubes: New Recreations,
Dover, 1981, 0-486-24140-8
A great book, but terminology and examples are not too clear. He refers
to 'perfect' and 'pandiagonal perfect' cubes
but these seem to be our definition for pandiagonal and pantriagonal
magic cubes.
Martin Gardner, Time
Travel and Other Mathematical Bewilderments, W. H. Freeman & Sons, 1988, 0-7167-1924-X
Includes a chapter on the first 'perfect' magic cubes, but
these are all by the old definition.
This is mostly from his Scientific America, Jan. 1976 column. He includes
references to some old works.
Andrews, W.S., Magic Squares & Cubes, Dover Publ., 1960 (original publication Open Court,1917)
The 'bible' on the subject of magic objects. Of course, it is very much
out of date but indicates how much was known 100 years ago.
Here is a more extensive bibliography of magic cube related literature.
.Please
send me Feedback about my Web site!
Harvey Heinz harveyheinz@shaw.ca
This page last updated
December 09, 2009
Copyright © 2002, 2003 by Harvey D. Heinz