Magic cubes are shown to have maximally symmetric inertia tensors if they are interpreted as rigid body mass distributions. This symmetry is due to their semi-magic property where each row, column, and pillar has the same mass sum. The moment of inertia depends only on this property and the number of point masses in each row, column, and pillar. Because magic cubes do not possess detailed cubic symmetry, other scenarios that result in maximally symmetric inertia tensors are discussed.
Topics
Rotational dynamics
REFERENCES
1.
Harvey D. Heinz and J. R. Hendricks, Magic Square Lexicon: Illustrated (HDH, Vancouver, 2000). Ordering information available from Harvey D. Heinz at [email protected] or 〈http://www.geocities.com/CapeCanaveral/Launchpad/4057/BookSale.htm〉.
2.
K.
Ollerenshaw
and H.
Bondi
, “Magic squares of order four
,” Philos. Trans. R. Soc. London, Ser. A
306
, 443
–532
(1982
).3.
Richard
Schroppel
in 1973 as reported by Martin Gardner, “Mathematical Games
,” Sci. Am.
234
, 118
–122
(1976
).4.
Harvey Gould and Jan Tobochnik, Introduction to Computer Simulation Methods: Applications to Physical Systems (Addison–Wesley, Reading, MA, 1996), 2nd ed.
5.
K.
Pinn
and C.
Wieczerkowski
, “Number of magic squares from parallel tempering Monte Carlo
,” Int. J. Mod. Phys. C
9
(4
), 541
–546
(1998
).6.
John R.
Hendricks
, “The third order magic cube complete
,” J. Recreational Math.
5
(1
), 43
–50
(1972
).7.
Walter Trump, 〈http://www.trump.de/magic-squares/magic-cubes/cubes-1.html〉.
8.
Peter D.
Loly
, “The invariance of the moment of inertia of magic squares
,” Math. Gaz.
88
(511
), 151
–153
(2004
).9.
Mary L. Boas, Mathematical Methods in the Physical Sciences (Wiley, New York, 1983), 2nd ed., pp. 413–425.
10.
Jerry B. Marion and Stephen T. Thornton, Classical Dynamics of Particles and Systems (Harcourt College Publishers, New York, 1995), 4th ed., pp. 405–423.
11.
Peter D. Loly, Frank Hruska, Marcus Steeds, and J. Williams, “Magic squares, linear algebra and eigensystems,” preprint 2003.
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© 2004 American Association of Physics Teachers.
2004
American Association of Physics Teachers
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