A curious walk through the Coordinate Plane

While I was doodling, I discovered something interesting regarding drawing a walk in the coordinate plane.

First, I selected 4 random lattice points as the vertices of a quadrilateral. Then, I started on a certain vertex. Next, I drew a line segment to the next vertex (going clockwise), then continued the line segment to have the extension be equal in length to the original line segment. Next, I did this again with the next vertex, drawing from where I was to the vertex, then drawing another line segment equal in length to my previous segment.

I noticed that whenever my original quadrilateral was a parallelogram, my path was bounded. However, if it wasn't a parallelogram, then my path became unbounded, growing indefinitely.

I tried the experiment again, except instead I chose a random point on the plane to start. Same results.

What do you guys think? Can you prove this conjecture? I managed to link this with another similar problem that is easy to prove. Can you find that problem?

#Quadrilaterals #Parallelogram #LatticePaths #CartesianCoordinates

Easy Math Editor

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Comments

Can you add a picture to go with your description? That will make following your instructions much easier.

Even an example in the case of a parallelogram could shed some light immediately as to why it would be bounded.

Calvin Lin Staff - 7 years, 5 months ago

It will be good if you post a picture of this as well :p .....

Eddie The Head - 7 years, 5 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$