Pentagons, Paper Folding, Stars & Origami

November 9, 2017

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/


 

11 Responses to “Pentagons, Paper Folding, Stars & Origami”

  1. goldenoj Says:

    Love this. Folding to 5 symmetry from a square is so intriguing, and the trig/geometry is sweet.

    Liked by 1 person


    • ..I just finished up figuring out a paperfolding solution to show another, non–trig way, to make that first fold that results in that nearly 72 degree angle that is essential to this folding. But am heading out of the house so I will have to file this with Fermat;’s last theorem for awhile…..

      Like

  2. Byopia Press Says:

    Thanks for this: best instructions ever for the four to five ratio! And of course I loved it because I have a thing about stars! ; ]

    Liked by 1 person

  3. goldenoj Says:

    The last ‘a little bit fiddly’ killed my star. But your pentagon works perfectly.

    Like


    • I haven’t checked this out yet, but I suspect that if you use my net of the star, and cut out those 5 little rhombi near the middle, that the “little bit of fiddly” will give you less trouble. Also, the thicker the paper, the harder it is to make that last bit work.

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      • goldenoj Says:

        Came back to it today. Part of my problem was misunderstanding the points fold, which finally I really looked at. The other thing was an idea that helped. The first point I folded I then unfolded so it could be the last point already half done.

        Liked by 1 person


      • Hey, I want some photo evidence of this! (please). I just got back to it today too, I tried out my thought of cutting out the little rhombi, and though it did make the folding easier, it also made it easier for the star to open up substantially in the middle…can be used as a cereal bowl now. Your solution to pre-fold is probably the best. Looking forward to pictures.

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  4. […] was reminded of it again recently when Paula Beardell Kreig wrote a blog post about discovering an instruction video for the stars. Paula got quite excited about the first part […]

    Liked by 1 person


  5. […] (@PaulaKrieg) explores some beautiful angle relationships in Pentagons, Paper Folding, Stars & Origami. And did you see her amazing crowd-sourced math art? Ta- Dah! I wish I’d remembered this post […]

    Like


  6. […] immediately when I saw Paul Beardell Krieg’s origami stars, I thought about turning one into a Santa Star. It occurred to me that the white pentagon formed on […]

    Liked by 1 person


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