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How do we put into words what we mean by a “square”? Here are six naughty shapes pretending to be squares. But they are all fakes! Can you see what’s wrong with each bad definition?
1) SQUARE: A SHAPE WITH FOUR SIDES OF EQUAL LENGTH: but hang on – this shape is not flat! It just goes around four edges of a cube. Hmm, let’s try another definition:
2) SQUARE: A FLAT SHAPE WITH FOUR SIDES OF EQUAL LENGTH: great, it’s flat! But surely a square has to have nice square corners? This shape does not! Well that’s easily fixed:
3) SQUARE: A FLAT SHAPE WITH FOUR SIDES OF EQUAL LENGTH AND FOUR RIGHT ANGLES: fantastic: these sides have lovely square 90⁰ Right Angles between them. That’s a square, right? But – oh no! – the ends don’t meet!! How about:
4) SQUARE: A FLAT CLOSED SHAPE WITH FOUR EQUAL SIDES AND FOUR RIGHT ANGLES: is this one a square?? Nope – because the sides aren’t straight – doh!
5) SQUARE: A CLOSED SHAPE MADE UP OF FOUR STRAIGHT SIDES. ALL THE SIDES HAVE TO BE THE SAME LENGTH: oh flip – I forgot about the right angles! Actually I didn’t, but I wanted you to see this – my favourite non-square. It’s called a skew cube. You can make one by joining four sticks at the corners, holding opposite sides and twisting them in opposite directions. See how the four sides actually make up four of the edges of a tetrahedron (triangle-based-pyramid). It’s very very cool – but it’s not a square.
6) SQUARE: A FLAT CLOSED SHAPE MADE UP OF FOUR EQUAL STRAIGHT SIDES: this one is nearly correct, but this shape of course fails to be a square because it doesn’t have 90⁰ corners. It’s still a very nice shape though: non-mathematicians call it a “diamond”, mathematicians call it a Rhombus.
So here at last is the correct definition:
What a mouthful – it’s a good thing we have a single word “square” to say all of that in one go!