I have chosen to write this page in narrative form.
The six sections each include a number of terms, which I show in boldface.
The meaning of these terms, if not specifically stated, can be inferred
from the contents.
Most of these terms are also be explained, using different wording, on my
perfect.htm and
perfect2.htm pages.
For a more comprehensive list of definitions for magic objects, go to my
glossary.htm page .
And for still more definitions, refer to the book Magic Square Lexicon:
Illustrated. [1]
Finally, for a very comprehensive list of concise definitions, see
Mitsutoshi Nakamura's
http://homepage2.nifty.com/googol/magcube/en/terms.htm.
These definitions for the most part, are consistent with those in the
above sources, but expressed in a logical, concise format and include many
tesseract classes as well. Especially note the section Conditions on
(pan)ragonals.
To simplify my explanations, I will use the general
term hypercube when referring to magic objects that may be a square, a
cube, a tesseract, etc.
I will use m to indicate the order of the
hypercube, and n to indicate the dimension.
[1] For information about the
Lexicon see booksale.htm.
Aspect
An aspect is an apparently different but in reality
only a disguised version of the magic square, cube, tesseract, etc. It is
obtained by rotations and/or reflections of the basic figure.
Once one has a hypercube of any dimension, through mirror images
(reflections) and rotations one can view the hypercube in many ways.
There are: A = (2n) n! ways of
viewing a hypercube of dimension n.
Dimension (n) 
Name 
Aspects 
2 
Square 
8 
3 
Cube 
48 
4 
tesseract 
384 

Any one of the aspects may be
considered the basic figure, or some criteria may be used to define
which is basic. For a magic square, the normalized position (basic
figure) is lowest corner number in top left corner, then lowest of
the two adjacent numbers to the right of it. 
I will illustrate aspect by first using a magic
square, then a magic cube. These examples of aspect will continue into the
section on associated.

This order3 magic square (fig.
a) is the smallest possible with primes in arithmetic progression.
[1] The second
magic square (fig. b) seems to be a different one. However, closer
inspection reveals that it is figure a reflected around a horizontal
axis. What was the top is now the bottom, and what was the bottom is
now the top. It turns out that there are 7 variations of the same
original magic square.
Normally, when counting magic hypercubes, the
aspects are not included, but are assumed to be variations of the
same hypercube.
Figure # 1  Basic is one of 4 basic cubes of
order 3. As mentioned above, each of these has 48 variations
(aspects) due to rotations and /or reflections. 

It is called a basic cube because
it cannot be transformed into another basic cube by rotations and
reflections.
Which of the 48 aspects is designated the basic one is rather
arbitrary. John Hendricks [2] defined the
standard position as the one with the lowest corner in the bottom
left position, and the 3 numbers adjacent to that corner in
increasing order in the x, y and z directions. Here I have shown
these four index numbers in red.The
second cube shown is a rotation and reflection of cube #1. Again I
have shown the four index numbers in red. However, obviously this
cube is not in the standard position.
In the next section, I show still another
aspect of this same magic cube. Notice that in all cases, the center
cell is the same. This will always be the case for odd order magic
objects that are different aspects of the same construction.
[1] H. E. Dudeney,
Amusements in Mathematics, Dover Publ. 1958, p. 246
[2] John R. Hendricks, The ThirdOrder Magic Cube Complete,
JRM 5:1:1972, pp 4350 
Associated
A magic hypercube where all pairs of cells
diametrically equidistant from the center of the hypercube equal the sum
of the first and last terms of the series, or m2 + 1 for a pure magic
square, is called associated. These number pairs are said to be
complementary. The series used is consecutive and starts at 1 in a normal
magic hypercube. However, even if the series consists of nonconsecutive
numbers, the complement of a particular number is found by subtracting it
from the sum of the first and last numbers in the series. This type of
magic square is often referred to as a symmetrical magic square.
All order 3 hypercubes are associated. The center
cell of odd order associated magic hypercubes is always equal to the
middle number of the series.
However, an odd order magic hypercube that is not associated, may also
have the middle number of the series in the center cell!
I will show some example associated magic squares,
then an associated magic cube.

Luoshu1 is the
smallest magic square and the only one of order 3 (not counting the
different aspects) and is associated. It uses the series of numbers
from 1 to 9. The middle number, 5, appears in the center of the
square. To obtain the complement
of each number, we subtract it from 10, which is equal to 1 + 9 (and
also m^{2} + 1).
We do this with each number in the square to obtain the
complementary magic square (Luoshu2). This is a different aspect of
the same Luoshu1 square and could also have been obtained by
rotating the original 180 degrees.
The fact that the compliment of the magic
object is a different aspect of the original is called
selfsimilar. This happens with all associated magic hypercubes
because they are centersymmetric. If the magic hypercube is
horizontalsymmetric (symmetric across a horizontal line) or
verticalsymmetric, the object is also selfsimilar.
Note that only centersymmetric hypercubes are called associated!
However, if the magic object is not
associated, or is not symmetric as described above, the resulting
magic hypercube will be a completely different one!
This order3 magic square (fig. c) is the
smallest possible with primes in arithmetic progression. While it
looks like a new magic square, it is actually a complement of figure
a in the section on aspects. So it is another example of a
selfsimilar magic square and is a 180 degree rotated aspect of the
original.
Because the numbers used are prime numbers
they obviously are not consecutive. The sum of the smallest number
and the largest number is 2078. Each of the four pairs of numbers
that surround the center of the square sum to 2078, making this an
associated magic square. We expected this because this is an order 3
hypercube!
If the magic hyper cube is an even order, it
may still be associated. In this case there is no center cell. The
complement pairs are diametrically opposite the center point
of the magic object.
This order 4 magic square (fig. d) is associated. The number series
used is 1 to 16. To complement this square we subtract each number
from 17 i.e. 16 + 1 (or m^{2}+1).
And again, we will obtain a different aspect, rotated from the
original by 180 degrees.
This cube is associated, because, as we stated
previously, all order 3 hypercubes are. However, that is confirmed
when we notice that the two numbers on each side of the center
number in all cases sum to 28 which is 3^{3} + 1.
The cube shown here is a third aspect of the cube shown in the
section on aspect. By subtracting each number from 28 (which is the
sum of 27 + 1), we obtain a fourth aspect of the same cube.
The central partitions of all odd order
associated magic hypercubes are themselves associated magic.
Examples:
The 3 central square arrays in an odd order magic cube are
associated magic squares.
The 4 central cube arrays in an odd order associated magic tesseract
(order 4 hypercube) are themselves associated magic cubes. Each of
these cubes, in turn, have 3 central associated magic squares.



Point of interest. Any of the four basic order 3
magic cubes can be reconstructed from just the four index numbers, because
the cubes are associated and contain only three numbers per line.
NOTE: To find out more about even order symmetric
(not associated) magic cubes see the self similar
page.
Constant (S)
The sum produced by each row, column, and nagonal (space diagonal)
of a magic hypercube (and possibly other arrangements). This value is also
called the magic sum.
The
order (m)
is the size of the hypercube. It indicates how many
cells
for holding numbers, that each line consists of.
For a
normal magic square, S = m(m^{2}+1)/2.
Minimum requirement is m rows, m columns and 2 diagonals
For a
normal magic cube, S = m(m^{3}+1)/2. Minimum
requirements for these classes.
Simple magic  m^{2} rows, m^{2}
columns, m^{2} pillars and 4 triagonals
Pantriagonal magic m^{2} rows, m^{2}
columns, m^{2} pillars and 4m^{2} pantriagonals
Diagonal magicm^{2} rows, m^{2}
columns, m^{2} pillars, 6m diagonals and 4 triagonals
Pantriagonal Diagonalm^{2}
rows, m^{2} columns, m^{2} pillars, 6m
diagonals and 4m^{2} pantriagonals
Pandiagonal magic m^{2} rows, m^{2}
columns, m^{2} pillars, 6m^{2} pandiagonals and 4
triagonals
Perfect magic m^{2} rows, m^{2}
columns, m^{2} pillars, 6m^{2} pandiagonals and 4m^{2}
pantriagonals
For a
tesseract (simple magic) S = m(m^{4}+1)/2.
Minimum requirement is m rows, m columns, m pillars, m
files and 8 quadragonals
In
general; for a ndimensional
simple magic hypercube S = m(m^{n}+1)/2.
Correct Summations and minimum
possible order for magic cubes Column 2
shows the minimum number of lines of m numbers summing to the constant.
There are often many other patterns of m numbers that also sum
correctly. 
Class 
Correct lines 
Order 
Simple magic 
3m^{2} + 4 
3 
Pantriagonal magic 
7m^{2} 
4 
Diagonal magic 
3m^{2} + 6m + 4 
6 
Pandiagonal magic 
9m^{2} + 4 
7 
PantriagDiag magic

7m^{2 }+ 6m 
8? 
Perfect magic 
13m^{2} 
8 

The above equations all assume the hypercube is
using a series of consecutive numbers that start with 1.
There is no reason this must be the case but it is so for a normal or true
magic square, cube, etc.
If the hypercube uses consecutive numbers that do
not start with 1. the equation is S = (mn+1) + m)/2 + m(a1),
where m = order, n = dimension, and a = starting
number.
Magic ratio
These terms were defined by Walter Trump in January,
2004. Their value is mainly for cubes that are almost magic.
They are also of value for cubes that are simple magic but not quite
diagonal magic (magic ratio).
Also for measuring magic cubes against a perfect cube (panmagic ratio).
Magic cube ratio:
The magic ratio is the number of correct
monagonals, diagonals, and triagonals
divided by the highest possible for a diagonal magic cube, which is 3m2
monagonals + 6m diagonals + 4 triagonals.
Panmagic cube ratio:
The panmagic ratio is the number of correct
monagonals, pandiagonals, and pantriagonals divided by the highest
possible, which is 3m^{2 }monagonals + 6m^{2}
pandiagonals + 4m^{2} pantriagonals.
Compact
and Complete
These are terms that are relatively new and have not
seen much usage to date. As we study and compare cube features, I will be
using these terms quite often.
I will describe the terms generally, then will
attempt to clarify the explanation with a diagram.
Compact:
This term was originally defined by Gahuko Abe
[1]
in 1990 as a feature of certain even order magic squares. Compact magic
squares consist of 2x2 subsquares where the 4 cells sum to S.
Kanji Setsuda [2] extended the
concept to magic cubes of order 4 where the four cells of all 2z2 squares,
parallel to the sides of the cube sum to 130 (which is S for order 4
cubes).
However, Setsuda used the term composite instead of
compact. Composite, though, is an old term for magic squares that are
composed from a group of small magic squares arranges as a magic square.
For that reason, I am reverting back to Abe’s
original term ‘compact’, to avoid confusion.
CompactPlus:
I have expanded on the concept of compact by proposing the term
compactplus for the following. If the corners of all subcubes of a
given order within the cube sum to 8S/m where m is the order of the
parent cube, then that cube is compactplus.
By this definition, an order 4 cube that is compact
may be considered compactplus because two parallel 2x2 squares, in fact,
form a 2x2x2 cube. The corners of that cube are 8*130/4.
However, compactplus is really meant for higher
evenorder cubes where all subcubes of one , or more, orders sum correctly.
For example, it seems that all perfect (nasik) cubes of order 8 have all subcubes
of all orders (2 to 8) correct. Not all higher evenorder nasik are
compactplus!
Because 'all' includes wraparound, there are m3 subcubes of each
order from 2 to m, contained within a cube.
Of course, because of wraparound, many of these m3 subcubes will
just be different arrangements of the same 8 numbers!
Complete:
Also defined by Kanji Setsuda [2]. To be complete,
every pantriagonal must contain m/2 complement pairs, and the two
members of each pair must be spaced m/2 apart. (Pointofinterest.
This is one requirement of a mostperfect magic square). As with the
compact cubes, the cube must be of even order. Also, the cube must be
pantriagonal or perfect (which implies pantriagonal).
Proper:
Refers to a cube that contains exactly the minimum requirements for that
class of cube. i.e. a proper simple or pantriagonal magic cube would
contain no magic squares, a proper diagonal magic cube would contain
exactly 3m plus 6 simple magic squares, etc.
NOTE that NO odd order associated magic hypercube can be proper because
all of these contain 3 central associated magic hypercubes of lower
dimensions!
This term was coined by Mitsutoshi Nakamura in an email of April 15, 2004
This example cube is pantriagonal, compact (and compactplus),
complete, and proper. 
Demonstration of compact.
Three 2x2 squares parallel to the cube faces
7, 50, 16, 57 are parallel to the sides
of the cube.
56, 41, 22, 11 are parallel to the
front of the cube.
49, 48, 25, 8 are parallel to the top
of the cube.
Demonstration of compactplus.
56, 41, 22, 11 and
1, 32, 35, 62 together are the 8
corners of a 2x2x2 subcube. These 8 corners sum to 260, which is
8/4 of 130. 
Demonstration of complete.
Because in the example cube, the number series used is 1 to 64, the
sum of the two complement members must be 1 + 64 = 65. I will show 3
example pantriagonals.
21, 2, 44, 63 is a 1 segment pantriagonal (a
main triagonal).
36, 59, 29, 6 is a 2 segment pantriagonal.
20, 55, 45, 10 is a 3 segment pantriagonal.
Demonstration of proper
Because no diagonals are correct, there are NO magic squares in
this cube, so it meets just the minimum requirements for a
pantriagonal magic cube. 
No order 3 are proper
No order 3 magic cube is proper because all contain magic squares. In
fact, this is true for order3 rectilinear objects of all higher
dimensions.
[1] Gakuho Abe, Fifty Problems of
Magic Squares, Self published 1950. Later republished in Discrete Math,
127, 1994, pp 313. The last 10 problems deal with magic cubes.
[2] Kanji Setsuda’s excellent Compact (composite) and Complete magic Cubes
Web pages may be accessed from.
http://homepage2.nifty.com/KanjiSetsuda/pages/EnglishP1.html
Cube parts
Rows, columns, pillars, files,
diagonals, pandiagonals, triagonals, pantriagonals, orthogonal
planes, diagonal (oblique) planes, broken diagonal planes.
For simplicity, I
am using an order 3 cube as an example (see fig. a). Order 3’s only
features are that it is simple and associated. Additional features
require higher orders.
Only the rows, columns, pillars and triagonals
are required to sum correctly for a simple magic cube. More parts
must sum correctly for the more advanced cubes.


Note that two diagonals go from corner to opposite
corner of each of the 3m orthogonal planes (2D figures). Four triagonals
go from corner, through the center, to opposite corner of the one cube
(3D figure).
Pandiagonals are the m1 twosegment lines that parallel each
diagonal.
Pantriagonals are the m^{2}1 two or threesegment lines
that parallel each triagonal.
Orthogonal planes
are those parallel to a face of the magic cube.
This is one of 3m such planes in a magic cube. Rows and columns must
sum correct for the cube to be magic. The 2 diagonals do not have to
be correct. However, if both are correct in a particular plane, then
that plane is a magic square. In this case they are, because the
plane shown is one of the 3 central planes of an associated magic
cube. As such, it must be an associated magic square. 

Diagonal planes (I usually call them
oblique planes) are those that go from 1 edge of the cube, through the
center, to the opposite edge. There are 6 of them joining the 12 edges of
the cube. In all 6 of these planes, both main diagonals must always be
correct. That is because the total of 12 diagonals are actually the 4 main
triagonals of the cube. In each plane, all rows or all columns must be
correct (because they are the rows or columns of the orthogonal planes. If
all rows and all columns in a particular plane are correct, then that
plane is a magic square.
Broken diagonal planes are the 2dimensional
equivalent of the broken diagonal pairs in a pandiagonal magic square.
There are m1 of these parallel to each of the six diagonal planes, The
only time these are of concern is in a perfect (nasik) magic cube. In that
case, each of these broken plane pairs automatically form a pandiagonal
magic square.
Classifications
This classification method is comprehensive,
coordinated, and may be consistently expanded for ever greater dimensions.
It was developed over the last 20 years or so by John Hendricks to avoid
the ambiguity over the term ‘perfect’ as applied to magic cubes.
[1]
Shown here are the 6 classes of dimension3
hypercubes, A reminder that there are only 2 classes of dimension2
hypercubes; the simple magic square and the pandiagonal (nasik) magic
square. There are 18 classes of dimension4 hypercubes (the magic
tesseract). They are listed here.
A. H. Frost published pantriagonal, pandiagonal and
perfect magic cubes in 1866 and 1878. He referred to them as Nasik cubes,
and did not differentiate between the three types.
[2].
Another early source on this subject is C. Planck's
Theory of Paths Nasik published in 1905. See a
direct quote from his paper at the
end of this page.
For more information on perfect magic cubes and some
history on the confusion over this name, go to my
cube_perfect.htm and cube_perfect2.htm
Simple magic:
m^{2} rows, m^{2}
columns, m^{2} pillars and 4 triagonals = S
Many simple magic cubes have some correct diagonals which means some magic
squares. Some of the oblique planes may have all correct rows and columns,
meaning that they also are magic squares. The diagonals of the oblique
planes are already correct, because they are actually the main triagonals
of the cube!
Mitsutoshi Nakamura suggested (April 15, 2004) the
use of the term 'proper' for a cube that contains only the minimum
features required for it's class.
A simple magic cube with NO magic squares for example, would thus be
referred to as a proper simple magic cube.
Pantriagonal magic:
m^{2} rows, m^{2}
columns, m^{2} pillars and 4m^{2}
pantriagonals = S Some pantriagonals
will consist of 2 segments and some will have 3 segments. (The four
main triagonals have only the one segment.)
Order 4 is the smallest pantriagonal magic cube
possible.
In the table I show the different pantriagonal cubes
I have examined for `associated` feature as of April 2004):

Order 
Number of different pantriagonal cubes seen 
Associated? 
4 
22 normal
2 (not consecutive numbers) 
No
Yes 
5 
6
2 
Yes
No 
6 
1 
No 
7 
4
4 
Yes
No 
8 
3 
No 
9 
2 
No 
10 
1 
No 
11 
2 
No 
12 
1 
No 
13 
1 
No 
14 
1 
Yes 
15 
1 
No 
16 
1 
No 
17 
1 
No 

Pantriagonal Diagonal:
m^{2} rows, m^{2}
columns, m^{2} pillars, 6m diagonals and 4m2
pantriagonals = S
The above sentence indicates that all 3m orthogonal arrays are
order m simple magic squares. All six oblique planes are also magic
squares.
In addition, all 4m^{2} pantriagonals must sum correctly.
This is a combination Pantriagonal and Diagonal magic cube.
This new class of magic cube was discovered by
Mitsutoshi Nakamura in 2004 and named by him. 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 known one being order 8 (and unassociated)
[5]. More about this on the Update3
page.
Pandiagonal magic:
m^{2} rows, m^{2}
columns, m^{2} pillars, 6m^{2} pandiagonals
and 4 triagonals = S
The above sentence indicates that all 3m orthogonal arrays are
order m pandiagonal magic squares. All six oblique planes are also
magic squares, usually simple, but it is possible that some of them are
pandiagonal.
Order 7 is the smallest pandiagonal magic cube
possible.
I have seen 12 of these cubes (Dec. 2003) All were order 7, and all but
one were associated.
In March , 2004, I received 7 more pandiagonal magic cubes, of odd orders
9 to 19. Five of these were not associated.
Perfect (nasik) magic:
m^{2} rows, m^{2}
columns, m^{2} pillars, 6m^{2} pandiagonals
and 4m^{2} pantriagonals = S
The above sentence indicates that all 3m orthogonal arrays are
order m pandiagonal magic squares. As well, all six oblique planes
are pandiagonal magic squares.
Furthermore, because all pantriagonals are correct,
there are m1 broken pandiagonal magic squares parallel to each of
the 6 oblique pandiagonal magic squares. [3][4].
Each of these broken magic squares consist of two segments. These oblique
planes may be transposed as horizontal or vertical planes to form a
different magic cube.
Order 8 is the smallest perfect magic cube possible.
Perfect magic cubes of orders 10, 14, 18, etc. and 12, 20, etc. are
impossible. this was proved by Rosser and Walker in 1939
[4], and by Stertenbrink and de Winkel in 2004. . See Pandiagonal
Impossibility
Proof.
Order 
8 
9 
11 
13 
15 
16 
17 
How many I have 
5 
4 
6 
4 
3 
2 
3 
How many are associated 
0 
3 
4 
1 
1 
1 
0 
Perfect magic cubes in my collection in
December 2005 
Above is Hendricks definition. Because of the
confusion over the definition of perfect, I suggest adopting the term
nasik. See my reason in The
Theory of Paths Nasik.
[1] H.
D. Heinz & J. R. Hendricks, A Unified Classification system for Magic
Cubes, JRM, 32:1, pp. 3036, 2003
[2] A. H.
Frost, Invention of Magic Cubes. Quarterly Journal of Mathematics, 7,
1866, pp 92102
A. H. Frost, On the General Properties of Nasik Cubes, QJM 15,
1878, pp 93123 plus plates 1 and 2.
[3] F. Liao, T. Katayama, K. Takaba, Technical Report 99021, School of
Informatics, Kyoto University, 1999. Available on the Internet at
http://www.amp.i.kyotou.ac.jp/tecrep/TR1999.html
[4] B. Rosser and R. J. Walker, Magic Squares: Published papers and Supplement,
1939, a bound volume at Cornell University, catalogued as QA 165 R82+pt.14.
[5] Mitsutoshi Nakamura has a good Web site at
http://homepage2.nifty.com/googol/magcube/en/
Classification of cubes by
included magic squares
A cube is considered 'proper' if it contains only
the minimum features to qualify for the classification (suggested by
Mitsutoshi Nakamura).
Simple magic cubes
May have some orthogonal or oblique magic squares, either simple or
pandiagonal.
Pantriagonal magic cubes
May have some orthogonal or oblique magic squares, either simple or
pandiagonal.
Only 2 of the 36 pantriagonal magic cubes of orders 4 to 11 that I tested,
had any oblique magic squares.
Diagonal magic cubes
All 3m orthogonal and 6 oblique planes are simple magic squares.
An order 8 diagonal magic cube, for example, contains 30 order 8 simple
magic squares. That is, 3m planar, and 6 oblique magic squares.
For a short time, I referred to this type of cube as a 'myers' cube. The
term 'diagonal', suggested by Aale de Winkel, is much more descriptive,
however.
On Nov. 13, 2003, Trump and Boyer discovered an order 5 diagonal magic
cube. It contains 21 simple magic squares. There are now diagonal cubes
for all orders greater then 4!
NOTE that Trump and Boyer refer to this class as a `perfect` cube.
Pantriagonal Diagonal
is a diagonal magic cube that is also a pantriagonal magic cube.
First discovered in late 2004 by Mitsutoshi Nakamura. the only known such
cube (Jan. 2005) is order 8.
Pandiagonal magic cubes
All 3m orthogonal are pandiagonal magic squares. The 6 oblique planes are
simple magic squares (several of these may be pandiagonal).
An order 7 pandiagonal magic cube, for example, the smallest possible,
contains 21 or 22 order 7 pandiagonal magic squares (and 6 or 5 simple
magic squares).
Perfect (nasik) magic cubes
Are a combination of pantriagonal and pandiagonal magic cube features.
All 3m orthogonal and 6m oblique planes are pandiagonal
magic squares. (All but 6 of the oblique planes consist of two segments)
An order 8 perfect magic cube, the smallest possible, contains 72 order 8
pandiagonal magic squares.
Order 9 is the smallest possible perfect associated
magic cube. The precedent for this is already set for perfect
(pandiagonal) magic squares. Order 4 is the smallest of these, but order 5
is the smallest pandiagonal associated.
Higher Dimensions
A tesseract is a 4dimensional hypercube. It is perfect (nasik) if
all panquadragonals 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 40m^{3}
lines that sum correctly. They are m^{3} rows, m^{3}
columns, m^{3} pillars, m^{3} files, 8m^{3}
quadragonals, 16m^{3} triagonals, and 12m^{3}
diagonals.
A magic hypercube of ANY
dimension n is perfect and
nasik if all panragonals sum correctly, and
all lower dimension hypercubes contained in it are perfect!
The r in panragonal indicates that we are talking about
all the agonals (orthogonal, diagonal, triagonal, quadragonal, etc). Pan
indicates all, i.e. main and broken.
Confusion over the term Perfect Magic Cube 
Historically, the term perfect has been used to describe any cube with
unusual features.
 In late 2003, Christian Boyer and Walter Trump
defined `perfect` to mean what I call a Diagonal magic cube. They
group the higher classes also under the term perfect, but with
enhancements.
 In late 2004, Mitsutoshi Nakamura adopted the
term Pan2,3agonal to replace the term perfect on his website. In this
way, he is avoiding the misunderstandings and ambiguities of the past.
 I, Harvey Heinz, will continue to support John
Hendricks definition of Perfect, and the coordinated system of magic
cube class definitions. I feel this system has real merit because it is
easy to understand, is logical, and is consistent for all dimensions and
orders of hypercubes.
 Recently, Heinz filled two gaps in Hendricks
coordinated hypercube definition by defining two new types of magic
cubes. It is now understood that as we go to higher dimensions, more
classes will appear. However, they will appear as gaps between the
recognized classes. The perfect class will always be the hypercube with
the highest number of features (and it's definition will not change).
By the end of 2005, this classification system is
gaining in popularity!
In the last few years, the term `nasik` is being used more frequently as a
substitute for Hendricks term `perfect`, in an effort to avoid confusion
(see the next section).
Theory
of Paths Nasik
Nasic, panragonal, and perfect.
Comments by Dr. C. Planck in 1917 (originally 1905)
on the features of a perfect magic hypercube.
This is a direct quote by Dr. C. Planck from W. S.
Andrews, Magic Squares and Cubes. 1917, Pages 365,366.
If the Nasik criterion be applied to
3dimensional magics what does it require? We must have 3 magic directions
through any cell parallel to the edges, (planar contact), 6 such
directions in the diagonals of square sections parallel to the faces
(linear contact), and 4 directions parallel to the great diagonals of the
cube (point contact), a total of 13 magic directions through every cell.
It has long been known that the smallest square which can be Nasik is of
order 4, or if the square is to be associated, (that is with every pair of
complementary numbers occupying cells which are equally displaced from the
center of the figure in opposite directions), then the smallest Nasik
order is 5. Frost stated definitely that in the case of a cube the
smallest Nasik order is 9. Arnoux was of the opinion that it would be 8,
though he failed to construct such a magic. It is only quite recently that
the present writer has shown that the smallest Nasik order in k dimensions
is always 2^{k}, (or 2^{k}+1 if we require association).
It is not difficult to perceive that if we push the
Nasik analogy to higher dimensions, the number of magic directions through any
cell of a kfold must be 1/2(3^{k}1), for we require magic directions
from every cell through each cell of the surrounding little kfold of order 3.
In a 4fold Nasik, therefore, there are 40 contiguous rectilinear summations
through any cell.
This subject is discussed in much greater detail in
[2]. In the introduction to that paper he says
Analogy suggests that in the higher dimensions we
ought to employ the term nasik as implying the existence of magic
summations parallel to any diagonal, and not restrict it to diagonals in
sections parallel to the plane faces. The term is used in this wider sense
throughout the present paper.
It must be pointed out here that Frost used the term
nasik for both his order 7 pandiagonal and order 8 pantriagonal magic
cubes in his 1866 paper. [1]
Planck, in the above quote is using it for the perfect magic cube
(which is a combination of the pantriagonal and pandiagonal cubes).
[2][3]
By Frost's definition:
 There is one nasik magic Square. Hendricks (and
many others) call it pandiagonal OR perfect.
 There are two nasik (Frost) magic cubes.
Hendricks called them pantriagonal and pandiagonal.
The combination of these 2 cubes is nasik (by Plancks definition)
Hendricks called it perfect.
 There are three nasik (Frost) magic tesseracts.
Hendricks called them pantriagonal, pandiagonal, and panquadragonal.
The combination of these 3 tesseracts is nasik (by Plancks definition)
Hendricks called it perfect.
(Actually, there are also tesseracts with
combination of two of the three panragonals. These are not
Nasik.)
To eliminate confusion between Hendricks and
Boyer/Trump 'perfect', I suggest Planck's term nasik for hypercubes where
all lines sum correctly.
 [1] A. H. Frost,
Invention of Magic Cubes, Quarterly Journal of Mathematics, 7,
1866, pages 92102. See page 99, para. 23 and page 100, para. 26.
[2] C. Planck, The Theory of Path
Nasiks, Printed for private circulation by
A. J. Lawrence, Printer, Rugby (England),1905
(Available from The University Library, Cambridge).
[3] W. S. Andrews, Magic Squares and Cubes. Open Court
Publ.,1917. Pages 365,366, by Dr. C. Planck.
Republished by Dover Publ., 1960 (no ISBN); Dover Publ.,
2000, 0486206580; Cosimo Classics, Inc., 2004, 1596050373
