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Squares of consecutive numbers differ by the sum of those numbers, so $6^2 = 5^2 + 5 + 6$. Geometrically, this is because the difference between the two squares is a pair of strips, $5$ and $6$ units long, that together form L-shaped polygon.

Is there a name for this L-shaped polygon? I vaguely recall that the ancient Greeks had a name for this shape, but I can't find it anywhere.

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The name for the shape is gnomon. From Wikipedia:

The ancient Greek mathematician and astronomer Oenopides used the phrase drawn gnomon-wise to describe a line drawn perpendicular to another. Later, the term was used for an L-shaped instrument like a steel square used to draw right angles. This shape may explain its use to describe a shape formed by cutting a smaller square from a larger one. Euclid extended the term to the plane figure formed by removing a similar parallelogram from a corner of a larger parallelogram. Indeed, the gnomon is the increment between two successive figurate numbers, including square and triangular numbers. The ancient Greek mathematician and engineer Hero of Alexandria defined a gnomon as that which, when added to an entity (number or shape), makes a new entity similar to the starting entity. In this sense Theon of Smyrna used it to describe a number which added to a polygonal number produces the next one of the same type. The most common use in this sense is an odd integer especially when seen as a figurate number between square numbers.

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  • $\begingroup$ Isn't that also the name of the part of a sundial that casts a shadow? If it is, I can see now why it's appropriate! $\endgroup$ – Phill Nov 30 '17 at 5:13
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It's a gnomon. Euclid used it often. Here's an example from Book II, Proposition 6:

enter image description here

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I usually call it "an L-shape". Most people would understand that, I think, and in the end that's what terminology is for: Not for discussing what's right and wrong to say, but for making yourself understood. Is that the actual word for it? I don't know. Does it really matter? I don't think so. If you feel uncertain, call it an L-shape, and give a brief description like you did here, or a drawing / figure, and you should be in the clear.

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  • $\begingroup$ I just have this vague recollection of hearing that the ancient Greeks had a name for this shape. I'll update the question to be more specific about this. $\endgroup$ – Doradus Nov 29 '17 at 13:55
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    $\begingroup$ An L shape includes any rectangle with a rectangle taken out of a corner. In fact it implies its not a square. $\endgroup$ – Harrichael Nov 29 '17 at 20:05
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    $\begingroup$ @Harrichael I agree that it doesn't imply squares. I do not agree that it implies not squares. $\endgroup$ – Arthur Nov 29 '17 at 21:18
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The name of the shape is a metaphorical name. In simple terms, a metaphorical shape is something that has its name from a really similar object in observable universe. Other examples being "Bell shaped curve", "cone", etc.

Reference link- Metaphorical Names

However, that is in common parlance, if you want to get a mathematical name for the same, it is- right-angled concave polygon wherein it can be a quadrilateral, hexagon or any other n-sided polygon such that n>=4

From the definition, a concave polygon should contain at least one reflex angle. A right angled polygon implies every angle has to be in odd multiple of 90 degrees, and adding concave would make it the exact definition of what we're looking for.

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