A Square of Your Own

January 21, 2018

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?

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/

Alison and her Milk Cartons

December 20, 2017

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.)

 

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/


 

To be two cubes

To be two cubes

I wanted to transfer this image to a big piece of paper. Way too big for my printer. It’s just under 24 square inches.

One the way to being two cubes

One the way to being two cubes

I made the pattern with the intention that it would fold into two cubes. BTW, I recently learned that the correct term to use here is net:” A pattern that you can cut and fold to make a model of a solid shape. This is a net of a cube.” (quoteth from the internet)

While I was scheming how to break the net into prints that I could piece back together, it occurred to me to just overlap the artboards in Illustrator. Set them up to be negative one inches apart. Here’s a snip of what the Illustrator workspace looked:

Six overlapping artboards in the Adobe Illustrator workspace

Six overlapping artboards in the Adobe Illustrator workspace

All I did, after setting up the six artboards was to overlay my net onto the artboards. No figuring, no scheming, just laid it right on top. Honestly I didn’t know what would happen. Would the overlapped parts not print? Just didn’t know.

Amazing. Everything printed everywhere. What I mean is that the parts of the image that were on the overlap printed on both papers. This made it really easy to piece together. Of course the best use of this technology is to print Happy Birthday banners. But what I did was piece them together, cover the back of the paper with blue crayon, and, using a ballpoint pen, trace over the lines to transfer to my larger paper.

net of the Cubes, cut out

I didn’t take any more photos of the process, but here’s my fully cut out net.

On the way to cubeness

The blue crayon showed up just enough, but what was really great is that the force of the tracing created score lines, making this easy to fold.

Weighted by a train

Weighted by a train

Here’s the cube. Hard to imagine how that image becomes these two two-inch cubes. So I made a video:

 

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