The Magic Encyclopedia ™

The Magic Object
(by Aale de Winkel)

The Magic Object is defined by this author as a general framework to put all possible magic objects, the loosely defined notational framework allows for most precise definition of an object in actual cases

Generalized Magic Object

General framework wherein all discussed item can be put

^mG-Magic Object {f(x_i)|g()}
m: The order of the object
with: G = IC, IR, IQ, IZ, IN
Object: Square, Cube, Tesseract ....
f(): number generating function
g(): 'magic condition'

The term "G-Magic Object" can in fact take any form, as well s the funvtions f() and g()

Magic Vector Space ^mIR-Magic Hypercube {"no restrictions"|"magic condition"}
The magic condition posed on a square of real numbers
viewed as vectors in a vector space

p-MultiMagic Hypercube _pⁿH_m ^mIN-Magic Hypercube {H[_ji] ε {1..mⁿ}| '_i=0∑^m (H[_ji])^k' = ^kt_n}

Magic Vector Space ³IR-Magic_Square

Viewing the order 3 magic squares as a vector space is quite interesting
using matrices as a vector space was new to me, so the reader more familiar with this
is invited to suggest other things

Base Like any vector space the space has a base the base below is normalised to 1.0 as the magic sum
B₀
1.0 0.0 0.0
-2/3 1/3 4/3
2/3 2/3 -1/3 B₁
0.0 1.0 0.0
1/3 1/3 1/3
2/3 -1/3 2/3 B₂
0.0 0.0 1.0
4/3 1/3 -2/3
-1/3 2/3 2/3
Note the transposed matrices of this base fall in the space
B₀^t = (1.0,-2/3,2/3) B₁^t = (0.0,1/3,2/3) B₂^t = (0.0,4/3,-1/3)

inner product
V . W A vector space define an innerproduct which is a number
note: that the below is defined on the matrices
V . W = V^i,jW_i,j = _i,j=0∑² V_i,jW_i,j

outer product
Cross product
V * W The cross-product defines a third vector based on two others
note: that the below is defined on the matrices
(V * W)_i,j = ((V_{(i+1)%3,(j+1)%3}W_{(i+2)%3,(j+2)%3}-V_{(i+2)%3,(j+1)%3}W_{(i+1)%3,(j+2)%3}) +
(W_{(i+1)%3,(j+1)%3}V_{(i+2)%3,(j+2)%3}-W_{(i+2)%3,(j+1)%3}V_{(i+1)%3,(j+2)%3})) / 2
This cross-product opposed to it's IR³ counterpart is symmetric V * W = W * V
Antisymmetric trials failed to remain within ³IR-Magic-Squares since the seperate parts aren't.

Magic Module ³IZ-Magic_Square

The Vector Space is defined on a Field, The Module has a simular definition on a Ring
so whereas the above numbers are on Fields like IR the below uses the field IZ

Base The following base is appropriate for ³IZ-magic_Square
IZB₀
0 1 -1
-1 0 1
1 -1 0 IZB₁
-1 1 0
1 0 -1
0 -1 1 IZB₂
1 1 1
1 1 1
1 1 1
The following base would be appropriate for ³IN-magic_Square
INB₀
1 2 0
0 1 2
2 0 1 INB₁
0 2 1
2 1 0
1 0 2 INB₂
1 1 1
1 1 1
1 1 1
Note; of course these bases can be expressed in the basis for ³IR-Magic_Square

inner product
V . W A vector space define an innerproduct which is a number
note: that the below is defined on the matrices
V . W = V^i,jW_i,j = _i,j=0∑² V_i,jW_i,j

outer product
Cross product
V * W The cross-product defines a third vector based on two others
note: that the below is defined on the matrices
(V * W)_i,j = 3 * ((V_{(i+1)%3,(j+1)%3}W_{(i+2)%3,(j+2)%3}-V_{(i+2)%3,(j+1)%3}W_{(i+1)%3,(j+2)%3}) +
(W_{(i+1)%3,(j+1)%3}V_{(i+2)%3,(j+2)%3}-W_{(i+2)%3,(j+1)%3}V_{(i+1)%3,(j+2)%3}))
Note this cross product is 6 times the one in ³IR-Magic_Square to avoid fractional
numbers in the resulting matrix, since the expresions doesn't stay in IN this defines only a
crossproduct in ³IZ-Magic_Square

Note: The above defines a weird situation, a few years back someone emailed me a question on how to use matrices as a Magic Vector Space, so I defined the above. The Cross-product I but recently defined and I sought for something anti-symmetric as it's counterpart in IR³ is defined I couldn't find expressions which remained in the vector space / module. The abve symmetric expression remained in the spaces. It remains to be seen what this all means. In IR³ the inner and cross product can be expressed in sin resp cos of the angle in between, thus one can define angles between vectors, the symmetric nature of the crossproduct might have interesting consequences.

I implemented this vector space in the 3IR-Magic-Square.zip spreadsheet. For the expert mathematician the academic question whether one can call the above defined "crossproduct" a "crossproduct", I know that in IR³ the crossproduct is anti symmetric, allowing one to define right-handed and left-handed coordinate system. Also the above relation to angles are taken from IR³, future investigation might reveal what this all meaans in ³IR-Magic_Square

Generalized Magic Object
General framework wherein all discussed item can be put ^mG-Magic Object {f(x_i)\|g()} m: The order of the object with: G = IC, IR, IQ, IZ, IN Object: Square, Cube, Tesseract .... f(): number generating function g(): 'magic condition' The term "G-Magic Object" can in fact take any form, as well s the funvtions f() and g()
Magic Vector Space	^mIR-Magic Hypercube {"no restrictions"\|"magic condition"} The magic condition posed on a square of real numbers viewed as vectors in a vector space
p-MultiMagic Hypercube _pⁿH_m	^mIN-Magic Hypercube {H[_ji] ε {1..mⁿ}\| '_i=0∑^m (H[_ji])^k' = ^kt_n}
Magic Vector Space ³IR-Magic_Square
Viewing the order 3 magic squares as a vector space is quite interesting using matrices as a vector space was new to me, so the reader more familiar with this is invited to suggest other things
Base	Like any vector space the space has a base the base below is normalised to 1.0 as the magic sum
	B₀ 1.0 0.0 0.0 -2/3 1/3 4/3 2/3 2/3 -1/3	B₁ 0.0 1.0 0.0 1/3 1/3 1/3 2/3 -1/3 2/3	B₂ 0.0 0.0 1.0 4/3 1/3 -2/3 -1/3 2/3 2/3
	Note the transposed matrices of this base fall in the space B₀^t = (1.0,-2/3,2/3) B₁^t = (0.0,1/3,2/3) B₂^t = (0.0,4/3,-1/3)
inner product V . W	A vector space define an innerproduct which is a number note: that the below is defined on the matrices
inner product V . W	V . W = V^i,jW_i,j = _i,j=0∑² V_i,jW_i,j
outer product Cross product V * W	The cross-product defines a third vector based on two others note: that the below is defined on the matrices
	(V * W)_i,j = ((V_{(i+1)%3,(j+1)%3}W_{(i+2)%3,(j+2)%3}-V_{(i+2)%3,(j+1)%3}W_{(i+1)%3,(j+2)%3}) + (W_{(i+1)%3,(j+1)%3}V_{(i+2)%3,(j+2)%3}-W_{(i+2)%3,(j+1)%3}V_{(i+1)%3,(j+2)%3})) / 2
	This cross-product opposed to it's IR³ counterpart is symmetric V * W = W * V Antisymmetric trials failed to remain within ³IR-Magic-Squares since the seperate parts aren't.
Magic Module ³IZ-Magic_Square
The Vector Space is defined on a Field, The Module has a simular definition on a Ring so whereas the above numbers are on Fields like IR the below uses the field IZ
Base	The following base is appropriate for ³IZ-magic_Square
	IZB₀ 0 1 -1 -1 0 1 1 -1 0	IZB₁ -1 1 0 1 0 -1 0 -1 1	IZB₂ 1 1 1 1 1 1 1 1 1
	The following base would be appropriate for ³IN-magic_Square
	INB₀ 1 2 0 0 1 2 2 0 1	INB₁ 0 2 1 2 1 0 1 0 2	INB₂ 1 1 1 1 1 1 1 1 1
	Note; of course these bases can be expressed in the basis for ³IR-Magic_Square
inner product V . W	A vector space define an innerproduct which is a number note: that the below is defined on the matrices
inner product V . W	V . W = V^i,jW_i,j = _i,j=0∑² V_i,jW_i,j
outer product Cross product V * W	The cross-product defines a third vector based on two others note: that the below is defined on the matrices
	(V * W)_i,j = 3 * ((V_{(i+1)%3,(j+1)%3}W_{(i+2)%3,(j+2)%3}-V_{(i+2)%3,(j+1)%3}W_{(i+1)%3,(j+2)%3}) + (W_{(i+1)%3,(j+1)%3}V_{(i+2)%3,(j+2)%3}-W_{(i+2)%3,(j+1)%3}V_{(i+1)%3,(j+2)%3}))
	Note this cross product is 6 times the one in ³IR-Magic_Square to avoid fractional numbers in the resulting matrix, since the expresions doesn't stay in IN this defines only a crossproduct in ³IZ-Magic_Square