Michael Stifel's Arithmetica Integra (1544) multiplicative magic square​
​by Oliver Seipel
sources: https://archive.org/details/bub_gb_fndPsRv08R0C
​https://archive.org/details/bub_gb_ywkW9hDd7IIC/page/n67/mode/2up​
​https://en.wikipedia.org/wiki/Michael_Stifel​
​https://en.wikipedia.org/wiki/Magic_square​
​https://en.wikipedia.org/wiki/Fr% C3 % A9nicle_standard _form
After constructing magic squares of any size Stifel gives two examples of “multiplicative” magic squares. They are related by the base-2-logarithm to the “additive” magic squares.
This tests for a magic square with operation Plus, Times, ...
In[]:=
MagicSquareQ[m_?SquareMatrixQ,op_:Plus]:=Equal@@Join[op@@@m,op@@@m,{Tr[m,op],Tr[Reverse/@m,op]}]
Testrun with Times:
In[]:=
MagicSquareQ[{{a,b,c},{d,e,f},{g,h,i}},Times]
Out[]=
abcdefghiabcdefghiaeiceg
Let’s check Stifel’s later example first:
In[]:=
Grid[Stifel3x3={{8,256,2},{4,16,64},{128,1,32}},FrameAll,BackgroundLightBlue]
Out[]=
8
256
2
4
16
64
128
1
32
Is this a multiplicative magic square?
In[]:=
MagicSquareQ[Stifel3x3,Times]
Out[]=
True
Now we take the base-2 logarithm:
In[]:=
Grid[Log2[Stifel3x3],FrameAll,BackgroundLightBlue]
Out[]=
3
8
1
2
4
6
7
0
5
You can add 1 for convenience:
In[]:=
Grid[1+Log2[Stifel3x3],FrameAll,BackgroundLightBlue]
Out[]=
4
9
2
3
5
7
8
1
6
Scholion: We all know, that up to rotation and reflection this is the only normal magic square of order 3.
Are these “additive” magic squares?
In[]:=
{MagicSquareQ[Log2[Stifel3x3]],MagicSquareQ[1+Log2[Stifel3x3]]}
Out[]=
{True,True}
Let’s check the first example now:
In[]:=
Grid[Stifel4x4={{32768,4,2,4096},{256,32,64,2048},{16,512,1024,128},{8,16384,8192,1}},FrameAll,BackgroundLightBlue]
Out[]=
32768
4
2
4096
256
32
64
2048
16
512
1024
128
8
16384
8192
1
Is this a multiplicative magic square?
In[]:=
MagicSquareQ[Stifel4x4,Times]
Out[]=
True
Now we take the base-2 logarithm:
In[]:=
Grid[Log2[Stifel4x4],FrameAll,BackgroundLightBlue]
Out[]=
15
2
1
12
8
5
6
11
4
9
10
7
3
14
13
0
You can add 1 for convenience:
In[]:=
Grid[1+Log2[Stifel4x4],FrameAll,BackgroundLightBlue]
Out[]=
16
3
2
13
9
6
7
12
5
10
11
8
4
15
14
1
Are these “additive” magic squares?
In[]:=
{MagicSquareQ[Log2[Stifel4x4]],MagicSquareQ[1+Log2[Stifel4x4]]}
Out[]=
{True,True}
Scholion: We all know, that up to rotation and reflection there are 880 normal magic square of order 4.