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Geometry/Symmetry of a Square

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Question
Dear Sir,

how many lines of symmetry a square has ?

Standard answer is four ! ("It can be folded in four different ways...") However, a square does not differentiate between two adjacent sides (with rotation for a Pi/2 around its center we arrive at the same square, not a different one). So, i opt for a TWO lines of symmetry.

(Any mention of "paper folding," "physical rotation," "x" and "y" axes etc. involves some additional, mathematical or physical structure, than a pure geometry.)

What is your opinion ?

with regards,
Dragan

Answer
Hi Dragan,

That's an interesting perspective.  Would you say a circle has only one axis of symmetry?

Most symmetries in the general sense do depend on some absolute frame of reference.  But that's also where applications of symmetry arise.  Although squares have only two different styles of axis of symmetry (corner-to-corner, and centre edge to opposite centre edge), the fact that there are two of each is relevant.  "It looks the same like this or like that" is, in some sense, the point of symmetry.  We can take advantage of the degrees of similarity, rather than neglecting them.

From your perspective, all regular polygons would have at most two axes of symmetry.
What 2D shapes would have more than two axes of symmetry?
An interesting discussion, to say the least.

Regards,
Azeem

Geometry

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Azeem Hussain

Expertise

I welcome your questions on algebra, 2D and 3D geometry, parabolic functions and conic sections, and any other mathematical queries you may have.

Experience

4 years as a drop-in and by-appointment tutor at Champlain College. Private tutor for dozens of clients over the past 8 years.

Publications
CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry

Education/Credentials
Bachelor of Science, Major Mathematics and Major Economics, McGill University, 2014. Diploma of Collegiate Studies; Pure and Applied Science, Champlain College Saint-Lambert, 2010.

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