Editing Frénicle standard form

A [[magic square]] is in the '''Frénicle standard form''', named for [[Bernard Frénicle de Bessy]], if the following two conditions hold:
# the element at position [1,1] (top left corner) is the smallest of the four corner elements; and
# the element at position [1,2] (top edge, second from left) is smaller than the element in [2,1].

In 1693, Frénicle described all the 880 essentially different order-4 magic squares.<ref>{{cite book |author=B. Frénicle de Bessy |display-authors=etal |title=Divers ouvrages de mathematique et de physique |year=1693}}</ref>

==Properties==
This standard form was devised since a magic square remains "essentially similar" if it is rotated or [[transpose]]d, or flipped so that the order of rows is reversed. There exist 8 different magic squares sharing one standard form. 
For example, the following magic squares are all essentially similar, with only the final square being in the Frénicle standard form:

  8 1 6   8 3 4     4 9 2   4 3 8     6 7 2   6 1 8     2 9 4   '''2 7 6'''
  3 5 7   1 5 9     3 5 7   9 5 1     1 5 9   7 5 3     7 5 3   '''9 5 1'''
  4 9 2   6 7 2     8 1 6   2 7 6     8 3 4   2 9 4     6 1 8   '''4 3 8'''

==Generalizations==
[[File:2009.10.23 quaternion square 01.png|thumb|right|384]]
For each collection of magic squares one might identify the corresponding group of [[automorphism]]s, the group of transformations preserving the special properties of this collection of magic squares. This way one can identify the number of different magic square [[equivalence class|equivalence classes]].

From the perspective of [[Galois theory]], the [[most-perfect magic square]]s (enumerated in {{OEIS2C|A051235}}) are not distinguishable since the size of the associated [[Galois group]] is 1.

== References ==
{{reflist}}

{{DEFAULTSORT:Frenicle Standard Form}}
[[Category:Magic squares]]