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The HEXAFLEXAGON -

a really neat folding paper gizmo!

Description, pictures, and how to order

 


Introduction

A hexaflexagon is paper disc that looks like it has two sides, a front and back, but when you "flex" it, the hexaflexagon reveals six (6) different faces. You will be amazed!

For example, here are the six faces of one of my own hexaflexagons.

Six faces of a hexaflexagon.
The six faces of a hexaflexagon.

 
Also, the flexing action is so therapeutic that you won't want to stop.

To be rigorously correct, these flexagons should be called hexahexaflexagons. That's worth knowing for web searches. The first "hexa" denotes the six different faces, and the second "hexa" denotes the shape of the flexagon. But the 6-faced job is the king of them all, so I simply call it the hexaflexagon. Heck, most of the time, I just call it a "flexagon". Accept no substitutes!

You can find other web pages which discuss the hexaflexagon. They will even tell you how to make one yourself. (See below.) But be forewarned - your first effort will take many hours, and your first flexagon will work just like the piece o' junk it's sure to be. Whoops, didn't mean to alienate anybody there!

Anyhow, why not buy one cheap from an experienced pro?

Now you can!

 


Hexaflexagon "Catalog"

Blank hexaflexagon

ITEM: Hexaflexagon
CATALOG NUMBER: dh1
ITEM DESCRIPTION: Hexaflexagon; 3.5 inches in diameter. Made of 20-lb. printer paper.

All prices shown are ALL INCLUSIVE. Prices include shipping and handling. See the "Word About Pricing" section at the bottom.

U.S. ORDERS:
1 Hexaflexagon (1 x dh1) = $ 9.39 (effective unit price = $ 9.39)
2 Hexaflexagons (2 x dh1) = $ 16.94 (effective unit price = $ 8.47)
3 Hexaflexagons (3 x dh1) = $ 24.51 (effective unit price = $ 8.17)
4 Hexaflexagons (4 x dh1) = $ 32.06 (effective unit price = $ 8.02)
5 Hexaflexagons (5 x dh1) = $ 39.62 (effective unit price = $ 7.92)
6 Hexaflexagons (6 x dh1) = $ 47.18 (effective unit price = $ 7.86)

FOREIGN ORDERS - CANADA and EUROPE:
1 Hexaflexagon (1 x dh1) = $ 10.12 (effective unit price = $ 10.12)
2 Hexaflexagons (2 x dh1) = $ 17.68 (effective unit price = $ 8.84)
3 Hexaflexagons (3 x dh1) = $ 25.24 (effective unit price = $ 8.42)
4 Hexaflexagons (4 x dh1) = $ 32.79 (effective unit price = $ 8.20)
5 Hexaflexagons (5 x dh1) = $ 40.35 (effective unit price = $ 8.07)
6 Hexaflexagons (6 x dh1) = $ 47.91 (effective unit price = $ 7.98)

Understand that your flexagons come blank. Let out your inner artist and decorate the faces in 6 completely different ways - with solid colors, colorful patterns, drawings, stickers, letters of a name, special messages, etc. I've had good luck with spray paint. You can even cut a photo into 6 equal pie slices and glue it onto a face. Anyhow, there's not much point to a flexagon if the faces all look the same.

Kids love them, too! (But don't pin me down on whether that's flexing the things, or destroying 'em!)

 


Ordering Instructions

Ordering is easy. Just follow these steps:

1. Check the address bar of your browser to make sure you are at www.donaldsauter.com/hexaflexagon.htm (with or without the "www"), which is the active and up-to-date version of this page. If not, go there now.

2. Send me an email at donaldsauter@gmail.com indicating how many hexaflexagons you want; the TOTAL COST; and your MAILING ADDRESS (optional, if your address on file with PayPal is up-to-date.) Nothing fancy; just use the following as a template:

    Hi Don, 

    I'd like 2 flexagons for $16.94.

    Sally
    SALLY GOODIN
    109 PEACH PIT DR
    COBBLER  GA  21901

3. Wait for a CONFIRMATION EMAIL indicating everything is in order. This is mainly for your benefit; I know I'm still in business, but you can't be sure this page is alive and well.

4. Deposit the TOTAL COST OF THE ORDER in my PAYPAL account, using my email address, donaldsauter@gmail.com . In PayPal, click the button for "This is a purchase of Goods." A short note in the "Email to recipient" box is optional; I usually have no trouble correlating the PayPal payment with the original request.

If you don't want to fool with PayPal, I can tell you how to send payment in the mail, no problem.

That's it. Sit back and wait a few days for delivery.

 


How To Make a Hexaflexagon

 
I won't attempt to explain it better than others have. Here, I think, are the clearest instructions on the web for making a hexaflexagon: Kathryn Huxtable's Instructions for Making Hexahexaflexagons.

 


How To Flex a Hexaflexagon

Kathryn also provides instructions on how to flex the hexaflexagon. Unfortunately, the pictures supplied with flexing instructions always seem lacking to me. Here is a sequence of photos which I hope will help.

The first step is just to look down on the flexagon lying on your two opened hands. Orient it so that one of the axes is going across, like from 9 to 3 on a clock face.

Flexing a hexaflexagon, step 1.
Flexing a hexaflexagon, Step 1.

Think of the index finger and thumb of your right hand as your "pincer fingers". Start your pincer fingers squeezing back the two triangular panels on the right side. At the same time, start the thumb of your left hand pushing in on the two panels on the left side.

Flexing a hexaflexagon, step 2.
Flexing a hexaflexagon, Step 2.

Complete both actions: squeeze the two right panels back against each other, and push the two left panels in so that, looking down on the flexagon, you see three "spokes" looking something like a T or Y, where each "spoke" is a pair of adjacent panels sqeezed together.

Flexing a hexaflexagon, step 3.
Flexing a hexaflexagon, Step 3.

This is very important:

BEFORE OPENING THE HEXAFLEXAGON TO REVEAL THE NEXT FACE, YOU MUST HAVE THE HEXAFLEXAGON FULLY SQUOZEN.

Keeping the flexagon squozen, reposition your fingers so that that your two thumb tips are moved up to the central top point.

Flexing a hexaflexagon, step 4.
Flexing a hexaflexagon, Step 4.

Now pull it open, carefully and uniformly. Imagine a blossom opening up.

Flexing a hexaflexagon, step 5.
Flexing a hexaflexagon, Step 5.

Below is a picture of what NOT to do. Do NOT open up one side of the flexagon separately before opening up the other. This may lead to panels from different faces getting mixed together. (I'm not really sure what people do to get a flexagon off-track.) And while there surely must be an inverse action to get the flexagon back on track, I've never been able to find it. To salvage somebody's off-track flexagon, I have to make a cut between two triangles, lay the strip out, refold it from scratch, and reattach where the cut was made.

What NOT to do (click to enlarge.)
What NOT to do 
(click to enlarge.)

Finish opening it up...

Flexing a hexaflexagon, step 6.
Flexing a hexaflexagon, Step 6.

...and you're ready for the next flex!

 

*** The 12 Flexes ***

Now that you have the flexing action mastered, you need to know how to get all 6 faces to appear. You will quickly find that it is not a simple matter of six flexes bringing out the six faces, one after the other. If you proceed in the most efficient manner, it takes 12 flexes to reveal all 6 faces and get you back to your starting point.

In constructing a hexaflexagon, I need to lightly pencil in a number, 1 to 6, on the front and back of each triangular panel in the original strip. "Face 1" of the completed flexagon, then, will show six 1's; "Face 2" will show six 2's; and so on.

You will erase these numbers before decorating your flexagon, but first, you might use them to practice getting all 6 faces to show up. I make flexagons according to the instructions given in Martin Gardner's column in "Scientific American" many years ago, and the completed flexagon has Face 1 backed with Face 2. Think of that as the "starting point".

For a flexagon with faces numbered that way, the most efficient route through all the faces requires a cycle of 12 flexes:

    Flexagon face:   1   3   6   1   3   2   4   3   2   1   5   2 
                     |                                   |
    Sequence starts here                             or here

Starting with the flexagon at the starting point, that is, looking down on Face 1 with Face 2 on the reverse, the next face you will reveal is either Face 3 or Face 5, depending on which axis you flex about.

Now, to proceed step-wise through the cycle, you must keep the flexagon in the same orientation throughout a flex. That is, do not add a rotating action to your flex. This may happen subtly, so you're not aware you're doing it. But while you're developing your flexagon "fingers" with the numbered faces, you can refer to the above cycle to keep you on track.

Suppose that Face 3 came up after your first flex. When you flex Face 3 and start to open it up, peek in to see whether Face 6 or Face 2 is ready to blossom. If it's Face 2, you added a rotation. Squeeze the flexagon back up, flatten it back out, rotate it a notch (to the next axis), and flex along that axis. Face 6 will appear.

On your next flex, if you follow these instructions of not rotating while flexing, you will find that the flexagon can not be opened up -- not to any face. You've hit a dead end. In that case, flatten it back out and rotate it a notch. Then flex to reveal Face 1.

Got it? Keep the flexagon in the same orientation as you flex, but back up and rotate a notch when you hit a dead end.

These instructions can't make much sense without a flexagon in hand, but, in any case, don't let them scare you off; you would eventually find all six faces by trial and error.

Did you git 'em all? Then erase those numbers and start decorating!

 


Gallery of Hexaflexagons

Debbie Hope's hexaflexagon.
Pretty art by my niece Debbie Hope; clunky designs probably by me.

 

Christmas card hexaflexagon.
Christmas hexaflexagon with stencil, speckly spray paint, happy guy,
paste-ons, red hat scheme, pie-sliced photo.

 


A Word About Pricing

First of all, sorry about the prices. I'm sure my "moves" are the most efficient anywhere, but it takes time to fold these things up!

The pricing is based on this: you pay for all the material and "shop costs" (just $.25), postage ($.49 First Class USPS), and the cut that paypal takes (2.9% plus $.30). I value my "skilled" labor in assembling the hexaflexagons at $40 per hour, and I figure a handling charge for each order, no matter how big, as 6 minutes at $10 per hour, which is $1. (Good deal?)

It takes me 76 minutes to construct 7 flexagons. That's 10.86 minutes (.181 hr) per flexagon. At $40/hr that works out to $7.24 for the labor in one flexagon. Everything on top of that is accounted for by the $.25 "shop cost", the postage, the paypal cut, and the $1 handling charge per order.

 


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