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One of the interesting myths about a certain building in our university is that it has 13 corners. One way to dismiss this claim is to point out that a polygon with right angles must have an even number of corners.

However, as pointed out in the linked page, this is not true.

Show that it is possible to construct a polygon that uses only right angles and has an odd number of corners.

(I understand that as a mathematical problem, this is understated in terms of what a polygon, right angle or a corner is. Feel free to define them in your own terms. I'd both like to see solutions that creatively define these terms as well as those which stick to the usual intuitive definitions. It'd be nice if your answer includes an explicit definition.)

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    $\begingroup$ How do you define 'polygon', exactly? Does it have to have a two-dimensional convex hull? Does it have to use Euclidean geometry? $\endgroup$ – Deusovi Oct 28 '19 at 22:44
  • $\begingroup$ @Deusovi Fair point. I don't have anything specific in mind. I have edited the question. $\endgroup$ – Agnishom Chattopadhyay Oct 28 '19 at 22:50
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    $\begingroup$ Then this question is likely opinion-based (as to whether something counts as a correct answer), no? $\endgroup$ – Deusovi Oct 28 '19 at 22:52
  • $\begingroup$ @Deusovi I'd rather call it an open-ended question. :) $\endgroup$ – Agnishom Chattopadhyay Oct 29 '19 at 2:41
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    $\begingroup$ @AgnishomChattopadhyay Both opinion-based questions and open-ended questions are largely considered off-topic on this site. $\endgroup$ – greenturtle3141 Oct 29 '19 at 4:28

10 Answers 10

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There are at least two possible ways to do this, depending on your definition of a polygon.

1:

Have the polygon in 3D space. For instance: start with an L shape, extend two bars from it upwards, and then draw a line connecting those two bars on a plane parallel to the L.

enter image description here
(The two dotted lines here are the ones that go "upwards".)

2:

Use hyperbolic or spherical geometry. Both of these are still two-dimensional, but they look distorted when projected onto a Euclidean plane. For instance, in this projection of the hyperbolic plane, each of these pentagon shapes actually contains five right angles and five straight edges, despite how they appear.

enter image description here
[Image source]

You can also draw a triangle with three 90-degree angles on a sphere: just go a quarter of the way around the equator, and then connect your start and end points to the north pole.

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  • $\begingroup$ Agreed, it's certainly usual to call them pentagons. $\endgroup$ – Paul Reynolds Oct 29 '19 at 13:35
  • $\begingroup$ For #1, I imagine an envelope with the flap open at a 90 to the body. I like it... $\endgroup$ – JeffC Oct 31 '19 at 20:54
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Lot of interesting answers here.

My attempt was this:

Admittedly, there are several pretty good definitions of corner which would not deem this as a solution.

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  • $\begingroup$ I think when speaking casually about a building, it would be quite reasonable to count the "double corner" as only one corner! And no exotic geometry is required. +1 $\endgroup$ – user253751 Oct 29 '19 at 12:33
  • $\begingroup$ Thanks, but I think that especially in the case of a building, it would count as two different corners. For example, you could have two different offices on two sides of the corners and they could be painted with different colors $\endgroup$ – Agnishom Chattopadhyay Oct 29 '19 at 14:22
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    $\begingroup$ Mathematically, maybe. Casually, "hey, did you know this corner is actually just the other side of the corner in Dr X's office? Neat!" $\endgroup$ – user253751 Oct 29 '19 at 15:01
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    $\begingroup$ I'd mark the wall with chalk, and walk around with my hand on the wall, counting corners as I pass. So this has 10 internal and 6 external. $\endgroup$ – Chris Cudmore Oct 29 '19 at 16:29
  • $\begingroup$ Where the 2 squares meet, I would have a tendency to think of that as 4 corners, not just 1. $\endgroup$ – computercarguy Oct 29 '19 at 17:42
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First of all, note that

with the usual definitions, it is in fact not possible to construct a polygon with only right angles and an odd number of corners. One way to see this: imagine walking around the polygon and keeping track of whether you have turned an odd or an even number of quarter-turns overall; when you get back to where you started you are facing the same direction as before and that number must therefore be even.

So

clearly we are looking for some unorthodox definition of something. The only really plausible candidate, I think, is "polygon". For instance, we might consider a polygon lying on some curved surface. Then our polygon would no longer need to be planar, and the argument above wouldn't work.

Here are two versions of this. First,

allow the edges of the polygon to be geodesics on whatever surface we have. In that case, there's a particularly simple solution: let the surface be a sphere, on which the geodesics are great circles, and then consider a spherical triangle with three right angles. (One way to make one: take three planes through the centre of the sphere all intersecting at right angles; these divide the sphere into 8 portions each of which is a spherical triangle with three right angles.)

But

you might find that unsatisfactory because those great circles aren't "really" straight lines, they're just "locally straight" within our curved surface. So, as an alternative, let's see whether we can construct a surface with enough actually straight lines on it to make this work. This is actually pretty easy. Let's construct the "polygon" first. In ordinary 3-dimensional euclidean space we'll start at (0,0,0) and then go to (1,0,0), then (1,1,0), then (0,1,1), then (0,0,1), then (0,0,0) again. Five straight-line sides, right angles between consecutive ones. Now for our surface. Start at the corners. Place a little square patch at each corner, in the plane spanned by the two edges meeting (at right angles) there. Then join adjacent patches with strips running along the edges, twisting a little as required. If you sketch this you will see that the result is a nice nonplanar pentagonal strip with a pentagonal "hole" inside it, and no weird nonorientable Moebius stuff going on. We can fill it in and extend it out pretty much however we like. The result is a surface on which those five straight line segments live happily, forming a right-angled pentagon.

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  • $\begingroup$ (Ninjaed by Deusovi, but only because I took the time to show how to turn my solution 2 (= his solution 1) into a polygon that actually lives on a surface, which I think makes it more reasonable to call it a "polygon".) $\endgroup$ – Gareth McCaughan Oct 28 '19 at 23:08
  • $\begingroup$ Wrapping triangles on spheres is hard work, and everything goes all bendy. Wrapping them on cubes is much more efficient :-) $\endgroup$ – Bass Oct 29 '19 at 9:37
  • $\begingroup$ When you say "imagine walking around the polygon ... when you get back...", you are assuming that the object is simply connected, aren't you? $\endgroup$ – Agnishom Chattopadhyay Oct 29 '19 at 14:40
  • $\begingroup$ I'm taking a polygon to be a chain of straight line segments. If you want to allow it to include multiple such chains then I remark that if each chain has an even number of vertices then so does their union -- aside from cases where some vertex is visited multiple times, as in your solution. (In that case of course you have to decide how to count vertices.) $\endgroup$ – Gareth McCaughan Oct 29 '19 at 16:05
  • $\begingroup$ @Bass yeah, but cubes are pointy and spheres are smooth. $\endgroup$ – Gareth McCaughan Oct 31 '19 at 11:05
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Following in the footsteps of @agnishom-chattopadhyay, here are a couple more ideas that rely on stretching the idea of what counts as a corner:

enter image description here

The polygon on the left is

a self-intersecting hexagon

so the midpoint counts as zero, one, two or four corners, entirely depending on your definition. (I don't think there's any reasonable way to count it as three, though.)

The polygon on the right (my favourite, and the reason I bothered to post this answer) has

partially coinciding sides, with a 90° turn in the coincident part.

The midpoint is a "double corner": going around the polygon, it serves as both a left and a right turn. This is (so far) the only 2D shape I see here that intuitively "feels like" it has an odd number of corners, but that is of course highly subjective, and there are many sensible definitions of "corner" that would cause the midpoint to be counted twice.

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One possibility is to simply have a

closed loop spiral.

In the attached image one could argue that the building, if so designed, fits that criteria even though the outside of the building has only four corners. This is a technicality, only, but still..

enter image description here

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Another solution is the answer to this age-old problem

A man walks 1 mile South, 1 mile East and 1 mile North and ends up back where he started, where is he?

Where part of the the answer is

He is at the North Pole

This makes a

triangle where each angle is 90 degrees

For completeness, the other half of the answer to that riddle is

that he is close enough to the South Pole that walking East for 1 mile takes him the whole way round the Earth at that point

A video explanation of both halves can be found here

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    $\begingroup$ The corner at the pole is not 90° (though you can fix that by adjusting the length of the eastward walk, or the size of the planet). The eastward portion isn't a straight line (it's actually a continuous left-turn for a quarter-circle), so calling the resulting shape a triangle is accurate only if you switch geometries. This is also pretty much identical with @Gareth's first answer. $\endgroup$ – Bass Oct 29 '19 at 9:30
  • $\begingroup$ Or @Deusovi's second. $\endgroup$ – Gareth McCaughan Oct 29 '19 at 16:05
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    $\begingroup$ This is easier to defend if you change the distances to 10000km. $\endgroup$ – Peter Taylor Oct 29 '19 at 17:08
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If the trickery is in the word "polygon" one other option is to make it

an open polygon

for example, a polygon

in the shape of an "L" would have 1 corner

but only have right angles

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The Oxford English Dictionary gives, for 'polygon', the following definition:

A plane figure with at least three straight sides and angles, and typically five or more.

This is similar to the definition given by Cambridge University. These definitions allow for answers not obviously possible when using the stricter, more mathematical definition found in e.g. Merriam-Webster's; it is not nesessary that a polygon have only straight sides, merely that it have at least 3 straight sides. A quarter circle, for example, counts as a polygon by this definition and has three right angle corners.

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What about this? It only has 7 or 9 right angles so it should meet your requirements.

enter image description here

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This reminds me of

The Numberphile Video on Youtube "5-Sided Square" which talks through the Non-Euclidian Geometry examples mentioned by @Deusovi of both a 5 sided and 3 sided 'square' (where square is defined as a shape with 90 degree corners).

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