folding · geometry and paper · geometry and paperfolding

A Square of Your Own

Squares, scaled
Squares, scaled

Making a plain old square exactly the size that you want can be such a chore. Too much can go wrong.

Squares can become so many different things (think ALL origami) that it’s worth knowing how to make a square just the size you want without getting stressed out.

Here’s how to do it, via the photo essay, followed by the video tutorial.

plain piece of copy paper
plain piece of copy paper

Here’s a regular piece of copy paper. The first steps look like I’m heading towards folding a square whose side is already predetermined by the short edge of the paper, but no. Be patient. This is going to be a surprise. Trust me.

First Step
First Step

Fold that short edge to meet the long edge, making sure that your fold ends (or starts) exactly at that corner. What you are doing here is bisecting that corner angle. If you cut away that rectangular flap on the edge, and unfold the triangle you’d have a square, but this isn’t what we are doing now. (Doing that was a different post, )

Marking the size of the new square
Marking the size of the new square

Unfold the fold then make a mark to indicate size of the square you want to make.

continuing...
continuing…

What I’m going to write next sounds horribly complicated but it’s easy to see and do.

Curl over that corner that has the fold going through it, lining up that corner point with the fold line it is leaning towards, and also lining up the upper edge of the paper with that mark you made. Press the fold.

Here’s a closer to look:

Closer Look
Closer Look

Now draw a line that traces around that folded down corner.

Time to cut out the square
Time to cut out the square

There’s your square!

Confused? Here’s a video to watch:

Got it?

Origami Pockets
Origami Pockets

Now make cool stuff.

Addendum: here’s a post that shows what I showed, plus another great way of making a square. https://mikesmathpage.wordpress.com/2018/01/23/a-fun-folding-exercise-for-kids-from-paula-beardell-krieg/

geometry and paper

Last Minute Wrapping Paper 2017

PaulaKrieg
Paula Krieg
Triangles

Last year, on December 24, I posted a page of decorative papers I had created to do some last minute wrapping. To make the papers I lifted images from pages and posts made by people whose work I love, so, naturally, I loved the papers. It was such a pleasure to do that I’m doing it again.

See the PDF at the bottom of the post to get the best copies.

Dan-Anderson's wrap
Dan Annderson
https://twitter.com/dandersod/status/939290160570789889

Here’s an exquisite doodle by Dan Anderson. Of this, he indicated that he didn’t really know what was going on with the creation of this graphic….which is how I usually feel about everything I do.

https://seekecho.blogspot.fr/2017/12/nicomachuss-theorem.html
Simon Gregg’s wrap
https://seekecho.blogspot.fr/2017/12/nicomachuss-theorem.html
https://twitter.com/mathhombre/status/940263419957178369

The page above is based off a graphic by Simon Gregg. A big part of the reason I picked this image is that it is part of a post that I really loved reading, https://seekecho.blogspot.fr/2017/12/nicomachuss-theorem.html

https://www.desmos.com/calculator/1vhiosg7gs
Suzanne von Oy
https://www.desmos.com/calculator/1vhiosg7gs

Suzanne Von Oy created a Desmos graph, which showed cows in a tornado. It was so silly that I, of course, had to play around with  it. That middle, green swirl in the picture above, is suppose to be kind of like a Christmas Tree with cow ornaments. (don’t judge me)

https://mikesmathpage.wordpress.com/2017/05/09/sharing-john-baezs-juggling-roots-post-with-kids-part-2/
Mike Lawler https://mikesmathpage.wordpress.com/2017/05/09/sharing-john-baezs-juggling-roots-post-with-kids-part-2/

Here’s another image that I chose based on how much I liked the post that it is part of. There were all sorts of wild shapes associated with this Lawler family post, but I like how rotating just one of the images made this fun pattern.

Malke Rosenfeld
https://mathinunexpectedspaces.wordpress.com/2017/12/10/star-o-rama-and-how-to-make-them/

When I saw this image stars-in-circles posted by Malke Rosenfeld, it just knocked my socks off.  Note the link in the caption, it says Star-o-Rama and how to make them!

https://twitter.com/GHSMaths/status/934026448767258624
Martin Holtham
https://twitter.com/GHSMaths/status/934026448767258624

It was unusually hard to figure out which of Martin Holthman mathart images to play with, as, lately, he has been having way too much fun making cool stuff. .Finally picked out this,  because it was not only awesome, but also seasonally snowflakish.

https://www.geogebra.org/m/xdHeBmYq
John Golden
https://www.geogebra.org/m/xdHeBmYq

I discovered the graphing program Desmos shortly after it came on-line. I’m not fluent enough in math to make amazing graphics with it (yet?).  Geogebra has been around for, well, I don’t know how long. I haven’t spent much time with it, but John Golden has spent a great deal of time with it. I really couldn’t decide which of his images to use, so I chose two. I considered chosing three, but I’ll save that third one for another project. I think that wrapping a small package with these undulating black and white lines will look really great. I’m especially looking forward to using this one.

https://www.geogebra.org/m/BqtGmER7
John Golden
https://www.geogebra.org/m/BqtGmER7

What’s the point of making mathy wrapping paper without some rhombic tori? No point at all. So here you have them, thanks to Mr. Golden.

If you want to print any or all of these papers,  I thin that the best way to do it is from a PDF.

Here’ the PDF Wrapping Paper 2017 . All nine images are here, but you can choose to print just the ones you need today using the dialog box on your printer.

Happy wrapping.

geometry and paper · geometry and paperfolding

Alison and her Milk Cartons

The Star that Nancy picked out to keep
The Star that my friend Nancy picked out to keep

The morning that I started this post I saw a series of photos tweeted out by Alison Martin. She’s been making some wondrous constructions using milk cartons. Here are two of the five tweets that caught my attention:

My attention was riveted on the fact that she had thought of this very accessible way to make slices of squares which she could then skew to make a veritable symphony of constructions. (Yes, am mixing metaphors. Will welcome edits…)

Sadly, our milk comes in glass bottles.

Also, since I do projects with students in schools, I have this thing about wanting to make everything from regular copy paper.

Paper Starbursts made from regular copy paper
Paper Starbursts made from regular copy paper

As I worked out how to create these starbursts I thought about the methods I use of working with paper that are not obvious. I made a video, in which I’m talking the whole way through, pointing out details of working with paper.

 

 

Here are some photos which repeat bits of what I’m showing in the video.

Not exactly a milk carton
Not exactly a milk carton, but it has a square end. Cuboid? Rectangular Prism?

Start with folding a tab. Next, ignoring the tab, fold a piece of regular copy paper into fourths, and then glue to the tab to make a shape with a square end.

Slice and glue
Slice and glue

Slice off a strip. Now, it’s not obvious how to continue, so don’t start slicing lots of strips. Just STOP after one slice.

Spread glue on one section of the tube (the part in the photo that has pencil lines on it)  then glue down that one slice of the paper at the edge of the tube. See photo below.

First Ray
First Ray, glued

Cut away another slice from the glued section, the same width as the slice that’s been glued on to it. Repeat and repeat…

A series of Rays
A series of Rays

…until you have 5 or 6 or 7 or 8 or however many rays you want or have materials for. I usually make 7 rays. Then glue the parts together that make them stay fully rotated.

The inside stars are a bit different to do. Make that shape with the square end again, but the width of the of this paper with should be about a third thinner than the paper you started with. (My first paper was 8 1/2″ wide, this second part was done with 5 1/2″ wide paper)

I’m going to let you figure out where the glue will go: this is pretty obvious.

What’s not so obvious is how to get the ray nicely placed.

What you do it this:

Placing an inner ray
Placing an inner ray

Splay open the big ray, slide in the small ray, then….(this is important, not obvious!)…

mvimg_20171220_171238~33729147328732120578..jpg
Sandwiching squeeze

...squeeze the neighboring rays together, which makes all layers align just right.

After making this inner layer of rays, if you want to make an even more inner layer of rays, don’t bother with making a closed shape with a tab, just fold paper into fourths.

mvimg_20171220_180530~22052134737897814684..jpg

See, like this.

mvimg_20171220_180947~33464398805300240078..jpg

Now here’s something I wasn’t expecting: they stand up by themselves.

And they stack.

(Watch the video. To the end. It’s much better than this post.)

 

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/