# Wonder what the scientific name for 4-sided symmetry is?

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Uprising
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11 Dec 2013, 2:53 pm

When a figure is mirrored both vertically and horizontally at the same time?

Stupid question maybe as google is failing on me with this and my dad has no clue either despite of him being an university graduate.

lelia
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11 Dec 2013, 5:04 pm

I don't know either, but that is a pretty image.

RushKing
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11 Dec 2013, 5:56 pm

It's D subscript 4

But if it had no reflections and 4 rotations, then it would be Z subscript 4.

Aoi
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11 Dec 2013, 7:45 pm

In geometric terms, the figure has is symmetric on rotation (in 90 degree or π/4 radian increments) and reflection (across axes that go from vertex to vertex, or from midpoint to midpoint of the edges).

In group theoretic terms, the figure has Z mod 4 symmetry for rotations, where Z refers to integers, and mod 4 means evenly divisible by 4. Without reflections, the figure can be represented by a dihedral group, again mod 4 with the same meaning.

Further symmetries can be found if this 2-D figure is assumed to exist in higher-dimensional space. If you image it on the x and y axes of a 3-d space, then it could also be rotated or reflected with respect to the z axis. Further extensions into higher dimensions are of course possible.

A good discussion of the symmetries of the platonic solids (this square-like shape is a 2-D projection of a cube) can be found in Artin or similar materials on abstract algebra.

To be fair to your university graduate father, this kind of question mostly comes up in abstract algebra or group theory, typically taught to first-year graduate students in mathematics, though often included in an undergraduate math curriculum.

TheWizardofCalculus
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16 Dec 2013, 10:45 pm

RushKing wrote:
It's D subscript 4

But if it had no reflections and 4 rotations, then it would be Z subscript 4.

Yeah. If it just was 2 reflectional symmetries, it would be:

Z_2 (semi-direct product) Z_2

It's been 5 years since my group theory class. 0o;

Fnord
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16 Dec 2013, 10:53 pm

"Four-fold Symmetry"