Harvey Heinz   Mass  Model   - 2001

 

 

 

 

 

 

 

 

(above illustration from Harvey Heinz web site)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Peter Loly – Moment of Inertia  2004

 

 

 

 

 

 

 

 

 

 

 

Finding the moment of inertia for the magic square

 

 

 

 

#1.  the unit distance between the center of two cells on x/y axis  = 1

 

#2.  the entire mass of the cell is presumed to be located at the exact center of the cell

 

#3.  the contribution of each cell to the moment of inertia for the square is given by the formula … (distance from center) * (distance from center) * mass of cell

 

#4.  the distance from the center of the square is of course found by the illustration below

 

 

 

 

 #5.  in the above illustration … finding the contribution of cell a1 to the moment of inertia is as follows …    

 a = 2 ,  b = 2  ,  c * c  =  8   …. 

 

Moment of inertia for cell  a1  =   8 *  (value in cell a1) .

 

 

 

 

 

 

figure above illustrates the  c *  c values for the inner shell

 

 

 

 

 

 

 

 

 

 

 

 

 

figure above  illustrates the  c *  c  values for the outer shell

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Francis Gaspalou     March 2008

 

 

 

                                                                                                                        March 08

 

GASPALOU’S PROOF

 

 

 

 

Theorem : in a 5x5 magic square

-         moment of inertia of the inner shell = 195-(center value*3)

-         moment of inertia of the outer shell = 1105+(center value*3)

Note : sum = 1300 (cf. Loly)

 

 

NOTATIONS

 

The cells of the magic square are called

 

A1 A2 A3 A4 A5

B1 B2 B3 B4 B5

C1 C2 C3 C4 C5

D1 D2 D3 D4 D5

E1 E2 E3 E4 E5

 

 

PARAMETERS OF A 5x5 MAGIC SQUARE

 

We can take for example the 14 parameters A, B, D, E, F, G, H, I, K, M, U, V, X, Y.

 

 

 

INNER SHELL

 

Moment of inertia = (B3 + D3 + C2 + C4) + (B2 + B4 + D2 + D4)*2

                              = (2*A + 2*E + M + 2*U + 2*Y - 65) + 2*(130 – A – E – 2*M – U – Y)

                              = 195 – 3*M

 

 

OUTER SHELL

 

Moment of inertia = (A1 + A5 + E1 + E5)*8 + (A2 + A4 + B1 + B5 + D1 + D5 + E2 + E4)*5 + (A3 + C1 + C5 + E3)*4

= 8*(A + E + U + Y) + 5*(65 + 3*M)

 + 4*(195 – 2*A - 2*E – 2*U – 2*Y - 3*M)

= 1105 + 3*M

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Walter Trump  - March 2008

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Moment of Inertia …. Order 7 Magic Square

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

*** Additional notes

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Looking at the patterns of the sum for each border in the order 7 magic square

In this example all numbers in outer border are identical

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Goofing around …