Regular Hexagon

Regular Hexagon

One of my all-time favorite paper moments was when I learned how to fold a regular hexagon. Many times, with  protractor or compass in hand,  I had tried to draw hexagons, but  they never worked out just right. This post, which features a tutorial page, is something that I have been wanting to do for a long time, but I needed to stumble upon just the right random instant of blog time. Recently, while musing about life, the universe and everything, well it seemed like the right time to finally put these drawings and steps down on paper. After all, the six goes neatly into 42.

Paper Folding and the Regular Hexagon by Paula Krieg

Paper Folding and the Regular Hexagon


Now, if you go ahead and make a hexagon for yourself, which of course you will because who could possibly resist trying this, you might notice a few splendid things. Then again, you might not notice them, so I will point them out.  First, you will notice that a preliminary step towards hexagonism is that you create an equilateral triangle , which is just the first of the many perks of this activity. The second, most extraordinary flash will be when you realize that the intersection of the three folded lines within your triangle is actually the center of the triangle. The reason that this is so remarkable is that this intersection point in no way looks like it’s at the center of the triangle. It just looks wrong as a center, and you might not believe it. But when you bring the tips of the triangle in to meet the intersection, well, let’s just say you will believe. 

Silently folding a hexagon drawn by Paula Krieg

Silently folding a hexagon

Just for fun, I decided to include this set of directions, too, because, really, it’s a much more attractive page than the one with all the writing on it. And there are plenty of people who will try it out without reading a thing, so here you have it.

Now after you’ve noticed that you’ve made a big equilateral triangle, there are few more shapes to uncover.  First of all, there are all sorts of little equilateral triangles inside of the hexagon. And if you fold the hexagon in half, well, you will have made an isosceles trapezoid. Now, think back, when is the last time you actually held an isosceles trapezoid in your hands? Next, fold back a third of the trapezoid, and there you have a rhombus. And if you can’t remember what any of these shapes are it’s probably because you never learned to spell them. Really, what it is it with this terminology? Wouldn’t these all these shapes be more memorable if we called them lollipops or kiwis?

A special nod here to Christopher Danielson. math teacherblogger who recently had way too much fun using hexagons in his classroom, so I wanted to add something to the virtual hexagon mix. And I want to acknowledge Steve Morris,  who kept me thinking way too long about the edges and shape of the universe: I no long think it’s purely a hexagon. That was just silly. Now I think it’s the shape that’s made when hexagons and pentagons are fitted together, but don’t be looking for a post about that. I think I need to get back to making books. Rectangular books.

An Actual Photograph of the Beginning of the Universe taken by Paula Krieg

An Actual Photograph of the Beginning of the Universe taken by Paula Krieg

I’ve been thinking about things that start in the middle. Since there’s been a good bit of talk in the news about the Theory of Inflation, aka theBig Bang, being validated I’m feel encouraged to be thinking about concentric expansions. In the middle of these thoughts, on cue, the phone rang. It was my cousin Pete, who calls me once or twice a year. Great! Just the right person!  I told him what I was thinking about. At first he seemed flummoxed, asking something like,”uh, what do mean? Give me an example.” I said, you know, like snowflakes, or the universe.

Snowflake captured by Kenneth G. Libbrecht using a specially designed snowflake photomicroscope.

Snowflake captured by Kenneth G. Libbrecht using a specially designed snowflake photomicroscope.

Of course, being Pete, he did know what I meant, and what he said next was really delightful. He said that things that start in the center generate other things that start in the center. I had been looking at antique lace doilies, so I could picture exactly what he meant.


I’m so used to thinking exclusively of things that have a linear path -a beginning, middle, and end – that coming up with examples of things that start in the center was, at first, challenging. But once the shift in thinking occurred  it was easier to come up things that followed this paradigm.  Flowers, for instance, start with a barely distinguishable bud…

Tiny bud and the back of an orchid flower

Tiny bud and the back of an orchid flower

…which expand and expand and then bloom.  Seeds, too, start from a central point, then send down roots,and send up shoots. And bowls, thrown on a wheel, start with a lump of clay and are formed through pushing out from the center.  And then there’s concentric ripples when a stone is thrown into a pond, and sound waves, too. I am still looking for examples so, when you read this, if anything comes to mind, please tell me in the comments what you’ve come up with.

What got me thinking about starting in the middle was something (hard to remember now…) about wanting to make a book that starts in the middle, and wanting it to be something about the number line…


…which really is not as visually compelling as a snowflake or an orchid. At least not yet.  Now, while I was quietly pondering all this, with my attention focused on the possible similarities between the universe and six-sided snowflakes my daughter called in from the kitchen asserting that six was a really perfect number. What? Was she reading my thoughts? No, she was making breakfast and admiring this cool contraption  that we have which submerges and times eggs for soft-boiling.

six-sided egg timer

Why, I asked her, was six such a good number? She said because it can balance two, three or four. That’s proof enough for me: so, yes, the universe must be a hexagon.

Mystery Book

March 13, 2014

Mystery Book 1

A friend of mine happened upon this little book some time ago. It’s about four inches high and just over an inch thick. It seems like it might be old, but maybe just a bit old. I don’t know. It’s housed in this sweet leather envelope. At least I think it’s leather. This is outside of my expertise. The sewing has the feeling of being old-fashioned gut thread, but, again, this is not something that I generally work with.

Mystery Book 2

The sewing on the cover piece has great personality. See the tiny little stitches on the bottom. I would say that the pages were sewn again the grain, as they are quite wavy, but they are stiff, so I am thinking that they are parchment or vellum?


I don’t recognize this writing at all. Seems like there’s some Greek letters here?


Some pages have a bit of red writing. There are some badly water damaged pages, too, where the writing has bled and smeared.

Mystery Book 5

The sewing is a link stitch. That much I know.


The covers are attached with a stitch through the wooden covers. I don’t know what kind of wood this is, but I suspect that a person who knows their woods would be able to identify this.  So, how about it? Anyone out there have any ideas on what to make of this little gem? I would be very interested in comments or musings, and I would be especially interested in knowing if the writing is just whimsical shapes, or if they are a recognizable language.

In snow again

In the meantime, my sun chair remains buried, nestled into yesterday’s fresh snow. It was 14 degrees Farenheit when I looked at the temp this morning. My daughter had the day off from school again, and I knocked myself out liberating my car from the icy mound that hid it. Before long we’ll all be up to our knees in mud, then it will be too warm. Oh well….hard to believe that just two days ago it was 60 degrees and absolutely glorious, one of the perfect days of the year.  I suppose they are all perfect days…after all, the snow storms keep my little family close to home.

drawing 1

Pencil on Layers of Tracing Paper

My daughter’s math class is working on logarithms. I have a special enthusiasm for logarithms. Every single thing about them appears to be overwhelmingly opaque and indecipherable.  Everything. And the most awesome thing about them being so completely crushingly incomprehensible is that Mr. John Napier (1550 – 1617) invented this system was to make life exponentially easier for us.  And he succeeded.

Now here’s another cool thing about logarithms. The spelling. No one confidently remembers how to spell this word. But there’s a trick to remembering.

How to spell logarithm

How to spell logarithm

The trick is to spelling logarithm is to notice that it starts with L O G (that’s the easy part) and ends with the most of arithmetic. No pun intended.

squares on page

Altered Graphs

I’ve been experiencing something that I mistook for an internal tug-of-war: I like blogging about book arts, but my mind of late has been drawn to playing with ideas that seem to have more to do with math than with books. It’s been a dilemma, how to keep writing about book arts when my mind is elsewhere. Finally I’ve had an ah-ha moment: I had forgotten that what brought me to book arts in the first place was wanting to make visual sequences of images that were related to a simple equation.


The fifth and the eighth page of a doubling book

The equation that drew me into making books is the one which starts with the number 2 and doubles, then doubles again, then doubles again and again.  It takes eight pages of doubling to get from 2 to 256. I’m infatuated by the slow measured way the numbers increase until there’s this tipping point, when  the quantities then erupt into unmanageable largeness.  I had created maybe a dozen of these books, experimenting with using lines, circles, overlapping lines, droplets of paint ect. I bound these books in a most inefficient and cumbersome way. Eventually a friend pointed me in the direction of The Center for Book Arts in NYC and new part of my education began. I found the  geometry of  constructing  books to be a satisfying, even sublime experience. And, since I didn’t really know any other intoxicating mathematical equations I just kept making books.


Solve for x                click here for the answer

Now, many years later, my daughter is coming home with problems like the one pictured above.  My son offered an insight on this kind of problem, one that I hadn’t thought of before. He said that the answer to this problem made no more sense to him that the problem itself.  It’s tough to remember that these functions have a look to them, and that solving for x looks like something.  A day or two after having this problem as part of a long, mind numbing homework assignment, my daughter came home and bemoaned that her teacher had just that day told them that the point of logarithms was to find exponents. She wanted to know why they weren’t told that in the first place, and what was the point of doing all those numerical gymnastics? We had quite the discussion about that. And it keeps me thinking about pictures.