2011 —
2014 —
See also other Log24 posts on quaternion group models.
Thursday, May 31, 2018
Quaternion Group Models
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Saturday, May 31, 2014
Quaternion Group Models:
The ninefold square, the eightfold cube, and monkeys.
For posts on the models above, see quaternion
in this journal. For the monkeys, see
"Nothing Is More Fun than a Hypercube of Monkeys,"
Evelyn Lamb's Scientific American weblog, May 19, 2014:
The Scientific American item is about the preprint
"The Quaternion Group as a Symmetry Group,"
by Vi Hart and Henry Segerman (April 26, 2014):
See also Finite Geometry and Physical Space.
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Thursday, June 5, 2014
Twisty Quaternion Symmetry
The previous post told how user58512 at math.stackexchange.com
sought in 2013 a geometric representation of Q8 , the quaternion group.
He ended up displaying an illustration that very possibly was drawn,
without any acknowledgement of its source, from my own work.
On the date that user58512 published that illustration, he further
pursued his March 1, 2013, goal of a “twisty” quaternion model.
On March 12, 2013, he suggested that the quaternion group might be
the symmetry group of the following twisty-cube coloring:
Illustration by Jim Belk
Here is part of a reply by Jim Belk from Nov. 11, 2013, elaborating on
that suggestion:
Belk argues that the colored cube is preserved under the group
of actions he describes. It is, however, also preserved under a
larger group. (Consider, say, rotation of the entire cube by 180
degrees about the center of any one of its checkered faces.) The
group Belk describes seems therefore to be a symmetry group,
not the symmetry group, of the colored cube.
I do not know if any combination puzzle has a coloring with
precisely the quaternion group as its symmetry group.
(Updated at 12:15 AM June 6 to point out the larger symmetry group
and delete a comment about an arXiv paper on quaternion group models.)
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Sunday, October 16, 2022
For Broomsday: Turning Eight
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Wednesday, June 4, 2014
Monkey Business
The title refers to a Scientific American weblog item
discussed here on May 31, 2014:
Some closely related material appeared here on
Dec. 30, 2011:
A version of the above quaternion actions appeared
at math.stackexchange.com on March 12, 2013:
"Is there a geometric realization of Quaternion group?" —
The above illustration, though neatly drawn, appeared under the
cloak of anonymity. No source was given for the illustrated group actions.
Possibly they stem from my Log24 posts or notes such as the Jan. 4, 2012,
note on quaternion actions at finitegeometry.org/sc (hence ultimately
from my note "GL(2,3) actions on a cube" of April 5, 1985).
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Monday, October 13, 2014
Raiders of the Lost Theorem
(Continued from Nov. 16, 2013.)
The 48 actions of GL(2,3) on a 3×3 array include the 8-element
quaternion group as a subgroup. This was illustrated in a Log24 post,
Hamilton’s Whirligig, of Jan. 5, 2006, and in a webpage whose
earliest version in the Internet Archive is from June 14, 2006.
One of these quaternion actions is pictured, without any reference
to quaternions, in a 2013 book by a Netherlands author whose
background in pure mathematics is apparently minimal:
In context (click to enlarge):
Update of later the same day —
Lee Sallows, Sept. 2011 foreword to Geometric Magic Squares —
“I first hit on the idea of a geometric magic square* in October 2001,**
and I sensed at once that I had penetrated some previously hidden portal
and was now standing on the threshold of a great adventure. It was going
to be like exploring Aladdin’s Cave. That there were treasures in the cave,
I was convinced, but how they were to be found was far from clear. The
concept of a geometric magic square is so simple that a child will grasp it
in a single glance. Ask a mathematician to create an actual specimen and
you may have a long wait before getting a response; such are the formidable
difficulties confronting the would-be constructor.”
* Defined by Sallows later in the book:
“Geometric or, less formally, geomagic is the term I use for
a magic square in which higher dimensional geometrical shapes
(or tiles or pieces ) may appear in the cells instead of numbers.”
** See some geometric matrices by Cullinane in a March 2001 webpage.
Earlier actual specimens — see Diamond Theory excerpts published in
February 1977 and a brief description of the original 1976 monograph:
“51 pp. on the symmetries & algebra of
matrices with geometric-figure entries.”
— Steven H. Cullinane, 1977 ad in
Notices of the American Mathematical Society
The recreational topic of “magic” squares is of little relevance
to my own interests— group actions on such matrices and the
matrices’ role as models of finite geometries.
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