On melancholic magic squares

Götz Trenkler; Dietrich Trenkler

Discussiones Mathematicae Probability and Statistics (2013)

  • Volume: 33, Issue: 1-2, page 111-119
  • ISSN: 1509-9423

Abstract

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Starting with Dürer's magic square which appears in the well-known copper plate engraving Melencolia we consider the class of melancholic magic squares. Each member of this class exhibits the same 86 patterns of Dürer's magic square and is magic again. Special attention is paid to the eigenstructure of melancholic magic squares, their group inverse and their Moore-Penrose inverse. It is seen how the patterns of the original Dürer square to a large extent are passed down also to the inverses of the melancholic magic squares.

How to cite

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Götz Trenkler, and Dietrich Trenkler. "On melancholic magic squares." Discussiones Mathematicae Probability and Statistics 33.1-2 (2013): 111-119. <http://eudml.org/doc/270964>.

@article{GötzTrenkler2013,
abstract = {Starting with Dürer's magic square which appears in the well-known copper plate engraving Melencolia we consider the class of melancholic magic squares. Each member of this class exhibits the same 86 patterns of Dürer's magic square and is magic again. Special attention is paid to the eigenstructure of melancholic magic squares, their group inverse and their Moore-Penrose inverse. It is seen how the patterns of the original Dürer square to a large extent are passed down also to the inverses of the melancholic magic squares.},
author = {Götz Trenkler, Dietrich Trenkler},
journal = {Discussiones Mathematicae Probability and Statistics},
keywords = {magic squares; patterns; group inverse; Moore-Penrose inverse; eigenvalues and eigenvectors},
language = {eng},
number = {1-2},
pages = {111-119},
title = {On melancholic magic squares},
url = {http://eudml.org/doc/270964},
volume = {33},
year = {2013},
}

TY - JOUR
AU - Götz Trenkler
AU - Dietrich Trenkler
TI - On melancholic magic squares
JO - Discussiones Mathematicae Probability and Statistics
PY - 2013
VL - 33
IS - 1-2
SP - 111
EP - 119
AB - Starting with Dürer's magic square which appears in the well-known copper plate engraving Melencolia we consider the class of melancholic magic squares. Each member of this class exhibits the same 86 patterns of Dürer's magic square and is magic again. Special attention is paid to the eigenstructure of melancholic magic squares, their group inverse and their Moore-Penrose inverse. It is seen how the patterns of the original Dürer square to a large extent are passed down also to the inverses of the melancholic magic squares.
LA - eng
KW - magic squares; patterns; group inverse; Moore-Penrose inverse; eigenvalues and eigenvectors
UR - http://eudml.org/doc/270964
ER -

References

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  1. [1] W.S. Andrews, Magic Squares and Cubes (Dover, New York, 1960). Zbl1003.05500
  2. [2] A. Ben-Israel and T.N.E. Greville, Generalized Inverses: Theory and Applications (Springer, 2nd edition, 2003). Zbl1026.15004
  3. [3] H.D. Heinz and J.R. Hendricks, Magic Square Lexicon: Illustrated, HDH, Surrey, BC, 2000. 
  4. [4] G.P.H. Styan, G. Trenkler and K.L. Chu, An Illustrated Introduction to Yantra Magic Squares and Agrippa-type Magic Squares, Lectures on Matrix and Graph Methods, Ed. by R.B. Bapat, S. Kirkland, U.M. Prasad and S. Puntanen (Manipal University Press, 2012), 159-220. 
  5. [5] D. Trenkler and G. Trenkler, Magic squares, melancholy and the Moore-Penrose inverse, Image 27 (2001), 3-9. Zbl1012.15005

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