The Windmill That Has No Mirror Line At All Symmetry

A flower, a butterfly, a rangoli, and a pinwheel all look strikingly balanced, while a photograph of clouds looks nothing like any of them -- because a symmetry is a part (or parts) of a figure that repeat in a definite, exact pattern, and clouds simply don't repeat that way. A rangoli's petals, for instance, come back to overlap themselves exactly whenever the whole flower is rotated by 90 degrees around its centre.

A line of symmetry is a line along which a figure can be folded so its two halves land exactly on top of each other -- confirmed directly by folding, not just by looking. A square, folded four different ways (straight down the middle vertically, straight across horizontally, and along each of its two diagonals), always produces a perfect match every time, giving it 4 separate lines of symmetry in total -- but an ordinary (non-square) rectangle's diagonal fold does NOT produce a match at all, since its two triangular halves are not mirror images of each other along that particular line.

Labelling a square's four corners and reflecting it across its vertical line of symmetry shows exactly which corners swap: the two corners on the fold line itself stay put, while the two corners on either side trade places with each other -- and reflecting the same square across a diagonal instead shows a completely different swapping pattern, with the two corners ON that diagonal staying fixed this time, and the other two trading places.

A windmill shape looks symmetrical at first glance, but trying to fold it along any line at all never produces a matching overlap -- it has no line of symmetry whatsoever. Rotating it by exactly 90 degrees about its central point, however, makes it look exactly the same as before -- and this is called an angle of symmetry, with the fixed point it turns around called the centre of rotation. The windmill's full set of angles of symmetry is 90, 180, 270, and 360 degrees -- and 360 degrees is always guaranteed to be an angle of symmetry for absolutely any figure at all, since a complete full turn brings everything back to exactly where it started, no matter the shape.

A figure built from three identical arms spreading out from a centre point, evenly spaced, has rotational symmetry exactly when the angle between each adjacent pair of arms is identical -- and since the three equal angles must together fill the complete 360 degree turn around the centre, each one must be exactly 360 divided by 3, which is 120 degrees. The same reasoning extends directly: five arms give 72 degree spacing, six arms give 60 degree spacing, and seven arms give a genuinely awkward result -- 360 divided by 7 comes out to 51 and 3-sevenths degrees, not a whole number at all, showing that not every radial arrangement produces a clean, whole-number angle of symmetry.

Every angle of symmetry a figure has always turns out to be a whole-number multiple of that figure's own smallest angle of symmetry -- since each further match is reached simply by rotating through the smallest angle an extra time. A circle takes this idea to its natural extreme: since a circle looks completely identical after ANY rotation at all around its centre, every single angle between 0 and 360 degrees counts as an angle of symmetry, and every diameter drawn through it is a line of symmetry too -- giving a circle infinitely many of both.