Geometric Drawings · geometry and paper · geometry and paperfolding · origami

Pentagons, Paper Folding, Stars & Origami

I came across a lovely way of folding stars. It was in a youtube video by someone named Tobias.

As lovely as these stars are, what really caught my attention was the way Tobias showed how to use paper folding to make a pentagon from a square. This square-to-pentagon transformation was in a separate video, and since it will take me about two days to forget everything I saw in the video I drew out the directions.

How to fold a Pentagon from a Square
How to fold a Pentagon from a Square. For the Video of this that Tobias made, go to https://www.youtube.com/watch?v=4kJmJUQVbO0

 

After the novelty (but not the thrill) wore off of making a pentagon from a square I began to look at the angles that I was making and figured that I could make the star with less steps (and perhaps with more accuracy) if I just started out with the net of the shape, so I made this map of the paper star’s fold lines:

Lines for a Folded Paper Star
Lines for a Folded Paper Star

If you make Tobias’s stars, after you get the hang of which lines fold in which direction, I highly recommend printing out lines above, score the lines with an inkless ink pen, and make that same star using just its essential folds.

The back of the paper sta
The back of the paper star

The photo above shows the backside of these stars. Quite a nice backside!

I’m sure that there are all sorts of things to do with pentagons, but something I want to mention is something that is fast and impressive, sort of the pentagon version of snowflake cuts. If you cut off an angled slice at the bottom of the folded up pentagon (step 12 in my tutorial drawing) there are all sorts of star possibilities.

36-54-90 triangles, with cutting lines on their tips
36-54-90 triangles, with cutting lines on their tips

These little beauties turn into:

Stars in Pentagons
Stars in Pentagons

The stars inscribed into these pentagons were made by cutting through all layers on the tips of the folded shapes.

 

And look, below there’s something extra for my friends who teach Geometry, and who might like a holiday themed angle activity. Part of the working out the folding pattern for the star was deciphering certain angles.

Find the Angles with degrees of 90, 45, and ~72, 18, 36, 54, and 108
Find the Angles with degrees of 90, 45, and ~72, 18, 23, 36, 54, 63 and 108

I had a good bit of help with the especially tricky parts of understanding the angle relationships. I’m sharing two twitter threads here, just because it was such a pleasure to get help from my friends.

and

That’s about it for now. Oh, and if you need to directions on how to fold a square from a rectangle, take a look at https://bookzoompa.wordpress.com/2014/12/10/paper-folding-squares-and-equilateral-triangles/


addendum March 2018

Here’s someone making this star. She makes it looks so easy! https://www.instagram.com/p/BfuSgYdnmY5/