Origami/Examples 1

FLYER edit

This is the "DanceWithStars.txt" file, which was 
created WED 2012 MAR 28 09:05 PM, 
revised SUN 2012 APR 08 10:32 PM.

Dance with the stars!
(even if you have two left feet and need a clock to keep time).
Hopefully, you will be able to sit in a comfortable chair 
next to a suitable work table. You won't even need to work up 
a sweat!

Dance with the stars -- Improved version -- no losers!
Everybody wins, and every winner takes home a handmade 
(made in USA) trophy -- or several trophies!


Teachers are especially invited! You will be able to share 
important, interesting, educational and cultural activities 
with your classes.

FREE! to the first ten people (any age above third grade) 
who sign up. After ten sign up, others will go on a waiting 
list for a possible follow-up session. Each meeting will 
consist of HANDS-ON activities; be prepared to have some good, 
clean fun! Be prepared to succeed in making something you've 
never before even imagined!

Paper and supplies will be provided.

An entertaining afternoon of unusual, but easy, craft projects 
is planned at the Caroline County Public Library in Greensboro. 
Mathematics only -- no arithmetic allowed, except by request. 
(Do you mean to tell me there is a difference between  
Mathematics and Arithmetic? I most certainly do! Come and find 
out what the difference is.)

Activities may include:
Origami (to fold paper)
Storigami (to tell a story, and illustrate it with origami)
Paper Sculpture
Paper Engineering 
and some other useful craft materials

---- ----- ----- ----- ----- ----- ----- ----- ----- -----

Above is a possible plan for the flyer, intended to attract 
attention, and to encourage people to sign up and attend.


Below is the plan of events. I hope that by posting this 
as a lesson plan at the Wikiversity I can establish some 
credibility for my outrageous claims. This should also allow 
me to post stories, pictures, and diagrams for participants 
to preview, download, and bring with. Of course, anyone on 
planet earth with access to the Wikiversity is free to use 
this material, once I post it.

Since by opening this project to the entire Greensboro 
community (to the entire world, via Wikiversity), I expect 
a wide range of ages, abilities, and prior experience, I 
plan to introduce some of the easiest, most fundamental 
crafts projects. Easy does NOT exclude four-dimensional 
geometry, vector calculus (without arithmetic, as much as 
possible), and discussion of non-orientable surfaces, and 
other topics as they arise. There are many strange, 
unusual, and unexpected things in Mathematics! Emphasis 
will be on hands-on, actual construction of interesting 
models. My goal is to make the "How to Make (Almost) 
Anything" course, popular at M. I. T., (though I have only 
read about it), accessible to a larger and younger audience.

NOTE: More activities must be planned than are expected to be 
actually used at any one event. Also, sometimes one must move 
on to another activity, due to lack of interest, or unexpected 
difficulties. Moreover, once a lesson plan is posted here, it 
is immediately available for anyone who wishes to use it. I 
intend to use the material myself, if there is ever another 
sequel event.

---- ----- ----- ----- ----- ----- ----- ----- ----- ----- 


Part of the beauty and wonder of origami is that NO TOOLS 
(other than the paper itself) are really required. However, 
some folders like to use the handle of an ordinary dinner 
knife to crease their folds. (If you fold a LOT or origami, 
your thumbnail may get uncomfortably HOT. And, you may wear 
grooves into your nails that interfere with other activities.) 
Scissors or a paper cutter are, of course, necessary to cut 
paper to specific sizes and shapes. (Even some authentic and 
historic Japanese origami sometimes requires cuts or slits 
in the paper.) Toothpicks, skewers, and tweezers are sometimes 
useful to put a stubborn flap into its proper place.

	Paper (almost all kinds -- except paper napkins)
	Knife with smooth handle (and no sharp blade)
	Paper cutter 
	Glue, tape, wire (These supplies are used mostly to 
		stabilize models for long-term display, so 
		they don't unfold themselves and look sloppy.)

---- ----- ----- ----- ----- ----- ----- ----- ----- -----



Inasmuch as not everyone has received the benefits of growing 
up in a family where construction engineering, drafting 
design, and other scientific and technological activities 
were everyday occurrences, I feel it to be necessary to give 
instructions for the preparation of the materials I expect to 

I am planning to bring prepared materials for these special 
projects, in order to allow you to begin working without delay.


Summary of Requirements for Popular Shapes, by shape and 
	by number required (see the source books)
Name			Shape	#	Shape	#	Shape	#
Regular Tetrahedron	3-sides	X 4 
Cube			4-sides	X 6 
Regular Octahedron	3-sides	X 8 
Regular Dodecahedron	5-sides	X 12 
Regular Icosahedron	3-sides	X 20

Truncated Tetrahedron	3-sides	X 4	6-sides	X 4
Truncated Cube		3-sides	X 8	8-sides	X 6 
Truncated Octahedron	4-sides	X 6	6-sides	X 8 
Truncated Dodecahedron	3-sides	X 20	10-sides X 12 
Truncated Icosahedron	5-sides	X 12	6-sides X 20
AKA Soccer ball, AKA Bucky ball 
Cuboctahedron		3-sides	X 8	4-sides	X 6 
Icosadodecahedron	3-sides	X 20	5-sides	X 12
Rhombicuboctahedron	3-sides	X 8	4-sides X 18 
Rhombitruncated		4-sides	X 12	6-sides	X 8	8-sides	X 6 
Rhombicosadodecahedron	3-sides	X 20	4-sides	X 30	5-sides	X 12 
Rhombitruncated		4-sides	X 30	6-sides	X 20	10-sides X 12 
Snub Cube		3-sides	X 32	4-sides	X 6 
Snub Dodecahedron	3-sides	X 80	5-sides X 12

Prisms			n-sides X 2	4-sides X n
Anti-Prisms		n-sides	X 2	3-sides	X 2 X n

Note: Some of these shapes are obviously precursors of the 
geodesic dome, invented by architect R. Buckminster Fuller, 
and featured in some museums of art. I did not make up the 
names of these shapes! But various (allegedly authoritative) 
sources sometimes get some of the names mixed up.


	Ordinary Poster Board (white, or colored, one or both 
		sides. What I get in Mayland, USA, is usually 
		22 inches by 28 inches.)

	Yardstick (or meter stick, if outside United States)
		( I can use my computer to convert 1 inch to 25.4 mm)
	Ball-point Pen, or pencil 

	Turn your measuring stick onto its edge when laying 
	out the dimensions. This puts the graduations on the 
	measuring stick closer to the poster board, which 
	should improve the accuracy of your lay out.

	Sight down along the edge of your straight-edged 
	measuring stick to be sure that it really is straight. 
	You want to draw straight line segments, not curves.

	The taper of a ball-point pen or a sharpened pencil, 
	when held against the straight-edge, leaves a small 
	gap between the edge of the straight instrument and 
	the location of the line which will be drawn. This 
	small gap is good; it helps prevent smearing of ink, 
	if you use a pen. But you will need to estimate the 
	size of this gap, carefully and accurately, in order 
	to draw the lines exactly where you want them to be.

	Clamp the straight-edge FIRMLY against the poster 
	board with one hand. You do NOT want it to move as 
	you are drawing the lines. With the pen or pencil 
	held in the other hand, far enough above the poster 
	board so as to leave NO MARK, practice a few times 
	making a smooth, sweeping stroke, while letting one 
	finger gently touch the guiding edge. After you gain 
	confidence that you can make a smooth, sweeping 
	stroke with a comfortable movement of your entire arm, 
	lower the pen or pencil to touch the poster board, 
	RELAX, and draw your first line. If that line looks 
	good: smooth, straight, and well-positioned, continue. 
	If it doesn't look so good, take a deep breath, try 
	to figure out what went wrong, reposition your 
	straightedge, clamp, RELAX, practice for a smooth 
	stroke, and try again.

	After a few years, all of these hints will become 
	automatic for a skilled draftsman. But you probably 
	want to help your kids with their homework THIS WEEK. 
	So, I try to share all these hints with you. Relax! 
	If you can cut the pieces accurate within a sixteenth 
	of an inch, you are doing very well, indeed.

	I have calculated (there goes my ban on arithmetic!) 
	dimensions to make many of the shapes, to the nearest 
	quarter of a sixteenth of an inch, in hopes of making 
	it easier for you to lay out these shapes.
	(I have also calculated the metric dimensions to use.)

	The same materials and tools are used for making all 
	of these shapes.

	Note: You will want to use regular poster board, that 
	you can easily cut with ordinary scissors. I am using 
	poster board, because I need to make a lot of these 
	shapes, and poster board (in various colors) looks a 
	lot nicer for public presentation than a wonderful 


	Salvage the cardboard from cereal boxes, snack boxes, 
	and other sources. Throw them away if they are 
	stained by garbage, but keep and use them if they are 
	clean and dry. Cut the boxes along the seams, so they 
	can lay flat for storage. The back side is usually a 
	plain gray or light brown color; ball-point pen ink 
	shows up well to mark your lines for cutting out shapes. 
	It's FREE, (or at least, already paid for), readily 
	available, and ecologically GREEN! Once upon a time 
	(decades ago, before TV became popular), packagers 
	would print designs intended for paper engineering 
	and paper sculpture on their boxes, to be cut out and 
	assembled into model cars, and trains, rockets, and 


	As you disect boxes to salvage the cardboard, you also 
	have an unusual opportunity to explore the many ways 
	professional packaging engineers have solved problems 
	important in manufacturing and commerce. Some boxes 
	are stapled together, some are die-cut, with flaps 
	and slots which lock tother, but most boxes are glued.

	Interesting problem: In how many ways can you unfold 
	a cubical box? Kunihiko Kasahara uses many of these 
	ways in his famous "Panorama Cube", as published in 
	_Origami_Omnibus:_Paper-folding_for_Everybody_, Japan 
	Publications, Inc., Tokyo, New York, {{ISBN|0-87040-699-X}},
	(paperback 384 pp.). This book also contains his 
	instructions for folding modules to make ALL of the 
	regular and semi-regular using only origami folding 

	Most boxes for commercial packaging are also printed, 
	often in may colors. You were probably taught in 
	school that the "primary colors" (for SUBTRACTIVE color 
	mixing) are red, yellow, and blue. These colors work 
	fairly well, for painting posters, but did you know 
	that professional printers usually use magenta, yellow, 
	cyan, and black inks? (OK. so they sometimes call 
	these colors "process red", "process yellow", 
	"process blue", and black. Still they refer to a "MYCK" 
	color system.) You will find their calibration marks 
	and the symbols used to align high speed printing 
	presses, printed in magenta, yellow, cyan, and black 
	ink on many boxes, if you look for them.

	Speaking of colors, your computer monitor uses a 
	different color system -- ADDITIVE color mixing. 
	The primary colors used for this are red, green, 
	and blue. Red and green, added together, make yellow, 
	which is the color of one of the inks used by printers. 
	Green and blue add to make cyan, and red and blue add 
	to make magenta. People who use computer monitors or 
	color television equipment often use these names for 

	Poster board (Do NOT try to use foam-core board for 
	these projects!)

	Yard stick
	Ball-point pen, or pencil


	Squares make a regular tessellation, so this should 
	be easy and obvious. I can usually trust the corners 
	of machine-made poster board to be accurate 
	90 degrees angles (i. e., square).

(1) Be sure to measure from the same short side each time when 
you make evenly spaced marks one inch (25.4 mm) apart along each 
long side of the poster board. 
(2) Be sure to measure from the same long side each time when 
you make evenly spaced marks one inch (25.4 mm) apart along each 
short side of the poster board.


	+------------+		+------------+	Do you see 
	|    |    |  |		|    |    |  |	the difference?
	|            |		|            |	Do you 
	|            |		|            |	understand 
	|    |    |  |		|  |    |    |	why it is 
	+------------+		+------------+	important?

The above instructions will be important for many other projects 
which require the laying out of grids.

	|  |  |  |  |  |
	+--+--+--+--+--+	You are trying
	|  |  |  |  |  |	to make a grid,
	+--+--+--+--+--+	something like
	|  |  |  |  |  |	this, only much 
	+--+--+--+--+--+	more extensive.
	|  |  |  |  |  |

(3) Connect all of the marks by drawing parallel line 
segments, as indicated in the diagrams above.
(4) Cut the poster board into strips along the lines you have 
(5) Cut each strip into squares.

	A piece of poster board 22 inches by 28 inches 
	should make 616 1-inch squares. 


	Although triangles and hexagons each make regular 
	tessellations, I prefer to use a semi-regular 
	tessellation which includes both shapes instead; 
	this makes it so much easier to cut out the hexagons.

	a            b                   f           d
	.            ________________________________.
	            /    \  /    \  /    \  /    \  /
	           /      \/      \/      \/      \/
	          /\      /\      /\      /\      /
	        /    \  /    \  /    \  /    \  /
	       /      \/      \/      \/      \/
	      /\      /\      /\      /\      /\
            /    \  /    \  /    \  /    \  /    \
	   /      \/      \/      \/      \/      \
	  /\      /\      /\      /\      /\      /\

	It is easy to make angles of 30 degrees and 
	60 degrees with a yardstick or ruler, when 
	you know how. In the diagram above, line 
	segment "bc" should be 6 inches long. To 
	construct this line segment, measure off 
	3 inches from the corner at "a".

	Note: This diagram is intended only to show 
	the princples of the construction! You will 
	achieve more accurate angles if you use longer 
	baselines. I suggest "ab" should be 12 inches 
	(or 30.48 centimeters), and "bc" should be 24 
	inches (or 60.96 centimeters). In any case, 
	you want the edges of the shapes you finally 
	cut out, to be 	one inch (or 2.54 centimeters) 

	Mark point "b" with a pen or pencil. Keeping 
	one of the graduations of the yardstick at 
	point "b", swing the yardstick (or meter stick) 
	until you find point "c", 6 inches away, at the 
	edge of the poster board. Now you can draw line 
	segment "bc". Mark off equal 1-inch (2.54 mm) 
	intervals along this line segment. Mark off 
	equal 1-inch (25.4 mm) intervals along the edge 
	"abfd" of the poster board. Measure length "ac", 
	then mark point "e" at that same distance from 
	edge "abfd". Now you can draw line segments 
	"ce" and "ef", then mark off equal 1-inch 
	(25.4 mm) intervals along each line segment. 
	This should give you enough grid points to 
	cover the poster board with triangles and 
	hexagons. Note: all lines should be parallel 
	to the edge "abfd", to line segment "bc", 
	or to line segment "ef".

	Once the grid is drawn, you can cut strips 
	of hexagons and triangles. Trim off all of 
	the triangles from each strip, and you 
	should be left with a pile of hexagons, and 
	another pile of equilateral triangles.

	I got [ ?? -- RCB ] hexagons and [ ?? -- RCB ] equilateral 
	triangles from my sheet of poster board. There 
	was some scrap near the edges of the sheet.


	Although the above polygons nest together to 
	form space-filling tessellations, regular 
	pentagons cannot fit closely together.

	There will be gaps between these and all of 
	the following shapes. I think that the 
	easiest way to make a lot of these shapes 
	is to first make a grid of carefully 
	calculated measured rectangles, then connect 
	the grid points with line segments which 
	outline the desired shapes. You have already 
	used this method once; since a network of 
	rectangles is also a network of squares, if 
	the dimensions are correct. I will do the 
	rest of the (ugh!) arithmetic for you, or 
	show you ways to avoid most of the arithmetic.

Example Calculations: 
	(You may skip down to "Procedures", if trigonometry 
	scares you. I'm trying to be a counter-terrorist, 
	myself.) There is an incredible irony here! I know 
	a very easy way to arrive at the measurements 
	without arithmetic, but I have a computer instead 
	of a drafting table and instruments. (Hey! I'll 
	keep the computer!) That very easy way (without 
	arithmetic) is simply to make a scale drawing 
	(using a protractor and a ruler) of a pentagon, 
	and measure the relevant dimensions.

	 |   _-  |\    |	This diagram of a regular 
	 |_-     | \   |	pentagon is about as good as
	B+-------|--\--+	I can make it in text mode.
	 |       |   \ |	Computer graphics is an
	 |       |    \|	enormously complicated 
	m+-----+-+-----+D	subject which I prefer 
	 |     O |f   /|	to postpone until some
	 |       |   / |	later time. 
	 |  -_   | /   |
	 |     -_|/    |	This is diagram one, 
	h+-------+-----+k	which will be mentioned 
	         E 		below.

	This diagram will serve for the purpose of
	being an example for the calculations,
	whose results follow. It also indicates 
	how the outline of the regular pentagon 
	will fit on the grid you will construct.

	We want line segment "AB" to be 1 inch 
	(or 2.54 centimeters metric). Point "O" is 
	supposed to be the center of our pentagon.	360 / 5 = 72
	Angle "AOB" is one fifth of a circle, or 72 
	degrees. Half of this angle is 36 degrees.	72 / 2 = 36 
	Let "m" be the midpoint of line segment "AB".
	Then angle "mOB" is 36 degrees, angle "BmO" 
	is 90 degrees, and line segment "mB" is 1/2 
	inch (or 12.7 mm, if you are using metric 
	90 18_-  |\54 90	Let's see if I can emphasize 
	 |_-  54|54\   |	the important angles.
	B+54    |   \36+
	 | -_ 72|    \ |	360/5 = 72
	 | 36-_|72  54\|
	m+-----+-------+D 	180 - 72 = 108
	 | 36_-O72  54/|
	 | _- 72|    / |
	A+_54   |   /36+	108 / 2 = 54
	 |72-_54|54/   |
	 90  18-_|/54 90	90 - 54 = 36

	Line segment "mOfD" is supposed to be a 
	horizontal diameter of the circle with 
	center "O", which passes through all
	five vertices of the desired regular 
	pentagon, and is a line of symmetry.

	This is enough information to apply 	Length("Bm")/Length("mO")
	elementary trigonometry (tri = three, 	= tangent(36 degrees)
	gono = angle, metry = measure), to 
	calculate other measurements of the 	Length("mO") =
	triangle "BOm".				Length("Bm")/tangent(36)

	Having emphasized the angles and the 
	triangles, (and having thouight for 
	several days about how best to provide 
	this information), I find that the 
	relevant facts are these:

	      H * cos(18)
	H *  |90     18_-
	sin  |      _-
	(18) |72_-  H

	(1) Right triangle "BCg" has acute angle 
	18 degrees and hypotenuse H = 1 inch 
	(25.4 mm). Elementary trigonometry (this 
	is what I was searching for) gives 
	Length("gC") = Length("BC") * cosine of 
	18 degreees. I also found that Length("Bg") 
	= Length("BC") * sin of 18 degrees.

	H * sin(18)
	  \      |
	   \     | H *
	    \    | cos
	   H \   | (18)
	       \ |

	(2) Right triangle "CDg" has acute angle 
	36 degrees and hypotenuse H = 1 inch
	(25.4 mm). Then Length("Cj") = Length("CD") 
	* sin (18 degrees), and Length("Dj") =
	Length("CD") * cosine(18 degrees).

	A quick little QB64 BASIC program gives 
	the measurement numbers we want to mark.


Pi = 4.0 * ATN(1.0)
' Computers and calculus students have an easier time calculating
' trigonometric functions when the angles are expresed in radians.
' Multiply the angle by Pi / 180 to convert degrees to radians.
PRINT "Pi ="; Pi

PRINT "This program calculates dimensions for grid to make regular pentagons. "
PRINT "16 * cos(36); 16 * sin(36):"
PRINT 16 * COS(36 * Pi / 180)
PRINT 16 * SIN(36 * Pi / 180)
PRINT "16 * cos(18); 16 * sin(18):"
PRINT 16 * COS(18 * Pi / 180)
PRINT 16 * SIN(18 * Pi / 180)
PRINT "(Dimensions in sixteenths of an inch.)"
PRINT "25.4 * cos(36); 25.4 * sin(36):"
PRINT 25.4 * COS(36 * Pi / 180)
PRINT 25.4 * SIN(36 * Pi / 180)
PRINT "25.4 * cos(18); 25.4 * sin(18):"
PRINT 25.4 * COS(18 * Pi / 180)
PRINT 25.4 * SIN(18 * Pi / 180)
PRINT "(Dimensions in millimeters.)"


Pi = 3.141593
This program calculates dimensions for grid to make regular pentagons.
16 * cos(36); 16 * sin(36):

16 * cos(18); 16 * sin(18):
(Dimensions in sixteenths of an inch.)

25.4 * cos(36); 25.4 * sin(36):

25.4 * cos(18); 25.4 * sin(18):
(Dimensions in millimeters.)

	But it was so much easier just to make 
	the scale drawing and measure off the
	dimensions I wanted! Here is a picture!


	Here is how to find the necessary dimensions
	(1) Tape a piece of paper to your drawing board.
	(2) Using your T-square pressed against the edge 
	of the drawing board as a guide, draw a horizontal 
	line near the middle of your paper. 
	(3) Using a drafting triangle, (pressed against the 
	T-square, which is still pressed against the edge 
	of your drafting table) as a guide, draw a second 
	line near the center of your paper, perpendicular to 
	the first line you drew.
	(4) Put the center of your protractor over the 
	intersection of the two lines. Align the 0 and 180 
	degree marks with the first line you drew on your 
	(5) Using the aligned protractor, put a mark at 72 
	degrees. Then, put another mark at 144 degrees. 
	(6) Turn the protractor 180 degrees, then re-align it. 
	(7) Make two more marks, at 72 and 144 degrees.	
	(8) Now, using a straightedge as a guide (a straight 
	side of your drafting triangle will do nicely), draw 
	four line segments which connect the marks you made
	with the protractor, to the intersection of the first 
	two lines you drew (Where the center of the protractor 
	(9) Make two marks 1/2 inch (12.7 mm) from the first 
	line you drew (one mark on each side.). 
	(10) Using your T-square as a guide, draw two new 
	lines, parallel to the first line you drew. These lines, 
	1 inch (25.4 mm) apart, establish the size, or scale, 
	of the regular pentagon you are constructing. They
	intersect the lines you drew at 144 degrees at points 
	"A" and "B" per diagram one.
	(11) Use your drafting compass to draw a circle through
	points "A" and "B", having its center at the 
	intersection of the first two lines you drew. 
	This will establish points "C", "D", and "E", 
	according to diagram one.
	(12) Now that you have located the five vertices of your 
	regular pentagon, connect them by drawing line segments 
	"AB", "BC", "CD", "DE", and "AE". 
	(13) Complete your drawing of the rectangle "gjkh" about 
	regular pentagon "ABCDE". Measure the parts of this 
	rectangle, then use these dimensions to lay out your 
	grid on your poster board.

	After you find the answer to a problem, 
	sometimes you wonder why it took you so 
	long to find the answer!

Information summary:

	What's in the diagram	US Measure 	Equivalent Metric measure

	Length("gB")		5/16 inch	7.85 mm   Measure off these
	Length("Bm")		8/16 inch	12.7 mm   distances along 
	Length("mA") = "Bm"	8/16 inch	12.7 mm	  one edge of your
	Length("hA") = "gB" 	5/16 inch	7.85 mm	  poster board.

	Length("gC")		15.25/16 inch	24.16 mm   Measure these along
	Length("Cj")		9.5/16 inch	14.92 mm   perpendicular edge.


	(1) Measure off and mark the four lengths "gB", "Bm", 
	"mA", and "hA" along one edge of your poster board. 
	(2) Repeat step (1), until you have marks all along 
	one edge of your poster board. 
	(3) Repeat steps (1) and (2) all along the opposite 
	edge of your poster board. 
	(4) Connect corresponding marks with a series of 
	parallel line segments. (I recommend using a long 
	straightedge to draw these lines.)
	(5) Two edges of your poster board have not been 
	marked yet. Along one of these edges, measure off and 
	mark the two lengths "gC" and "Cj". 
	(6) Continue measuring and marking lengths "gC" and "Cj",
	all along the edge you have started marking. 
	(7) One edge of your poster board has not been marked yet. 
	Use lengths "gC" and "Cj" to mark this edge.
	(8) Connect corresponding marks with a series of 
	parallel line segments. (I recommend using a long 
	straightedge to draw these lines.) 
	(9) Use a short straightedge as a guide to draw all five 
	sides of each regular pentagon in your grid.
	(10) Cut your poster board into strips, so that each 
	strip contains an entire row of regular pentagons. 
	(11) Cut each strip into rectangles, so that each 
	rectangle contains a regular pentagon.
	(12) Trim each rectangle. Keep all of the regular 
	pentagons. Discard all of the triangular scraps.

	I got [ ?? -- RCB ] regular pentagons from my sheet of 
	poster board. Some scrap had to be trimmed from each 
Regular Pentagon layout

Ray Calvin Baker (talk) 03:14, 15 May 2012 (UTC)


	+m    +-------+    n+	Note: This diagram is
	     /C       D\     	distorted. (It's too
	    /           \    	tall.) Technical 
	   /             \   	difficulties such as
	  /               \  	this often arise when 
	 /                 \ 	one tries to push
	+B                 E+	equipment beyond the
	|                   |	limits for which it
	|                   |	 was designed . Word 
	|                   |	processors were never 
	|         O         |	designed for making 
	|                   |	diagrams. But creative 
	|                   |	thinking often requires 
	|                   |	that one thinks beyond 
	+A                 F+	the normal limits. 
	 \                 /
	  \               /  	This shape has four-fold 
           \             /   	rotational symmetry, so 
	    \           /    	a lot of the lengths in 
	     \H       G/	the diagram are identical.
	+q    +-------+    p+

	Angle "mBC" is supposed to be 45 degrees. A true 
	scale diagram, or trigonometric calculation, would 
	establish this as a fact. Triangle "mBC" is thus 
	an iscoceles right triangle, with some interesting 
	and unusaul properties. If Length("BC") = 1 inch 
	(25.4 mm) then length("mC") = length("mB") = 
	cosine(45 degrees) = sine(45 degrees) = 1/2 the 
	square root of 2 = 0.7071.

Table of measurements:
	Length("mB") = Length("mC")	11.25/16 inch	(17.96 mm)
	Length("AB") = Length("CD")	1 inch		(25.4  mm)
	Length("qA") = Length("Dn")	11.25/16 inch	(17.96 mm)

	(1) Measure off the dimensions for one cell of the grid
	along one edge of your poster board. 
	(2) copy these measurements along the edge to make as 
	many grid cells as possible along that edge. 
	(3) Repeat steps (1) and (2) along each of the other 
	three edges of your poster board. (Remember to start 
	all of your measurements from the correct edge of the 
	poster board.) 
	(4) Use a long straightedge as a guide to draw line
	segments connecting corresponding measured marks.
	(5) Cut your poster board into strips along the 
	grid lines you have drawn. 
	(6) Cut each strip into squares along the grid lines. 
	(7) Trim away the triangles from each square. 
	(8) Discard the triangular scraps.

	Four triangles of scrap had to be trimmed from each square 
	to make [ ?? -- RCB ] regular octagons -- "stop signs".


	The computations for laying out the grid for this shape 
	are somehat like the process for laying out the regular 
	pentagons, except for the essential fact that there are 
	twice as many sides for this 10-sided shape.

Table of measurements:





	None of the regular or semi-regular polyhedra 
	require this shape, but it can be used nicely to 
	make a pretty semi-regular tessellation, prism, 
	or anti-prism, so I try to include a few instances 
	of this shape.

Table of measurements:





	The special tool you will want for this project is 
	a loop of wire which will fit through a drinking 
	straw, to pull a length of string through the straw.

	Wire (light gauge doorbell hookup wire from a 
		hardware store works just fine)

	Wire cutters
	Pliers (needle-nosed pliers work best)
	Ball-point pen or pencil

	(1) Measure off a piece of bell wire about 50 per cent 
	longer than a drinking straw. (Drat! More of that 
	(2) Form a loop at each end of the piece of wire, 
	using the needle-nosed pliers. DON'T POKE YOUR EYE OUT! 
	To minimize the danger of that, I recomment a loop 
	at each end of the piece of wire.
	(3) Twist the short end of the loop around the wire 
	several times. Do this with each of the two loops. 
	(4) Squeeze each loop down to size, so that it will 
	fit easily through the drinking straws, while keeping 
	the loop large enough to slip a piece of string through it. 


	Take one sheet of paper 8+1/2 inches by 11 inches for 
	each star you wish to make.

	Ball point pen or pencil

	The end of a ruler or yardstick sometimes gets battered 
	and worn, and may not be well aligned with the graduations
	of the measuring instrument. To avoid these possible 
	errors, I usually align the 1-inch mark with the place from 
	which I wish to measure. This can cause its own type of 
	errors, but it is usually easy to spot and fix if your
	measurements are off by exactly one inch.

	(1) Align your ruler with the 1 inch mark at the edge of the
	paper. Mark along both of the short edges at 3 inches, 
	at 5 inches, at 7 inches, and at 9 inches. This will 
	leave 1/2 inch of waste along the long edge. 
	(2) Align your ruler with the 1 inch mark at the edge of the
	paper. Measure and mark along the long edges at 4+1/2 
	inches, at 8 inches, and at 11+1/2 inches. (I avoid using 
	a hyphen in mixed numbers like these; it can too easily be 
	mistaken for a "minus" sign, leading to subtraction instead 
	of addition.) This process of measuring and marking will 
	leave 1/2 inch of waste along the short edge. 
	(3) Using the ruler as a straight-edge, draw line 
	segments to connect the marks. There should be four 
	lines running the long way, and three lines running 
	the short way. 
	(4) Cut the paper along the lines.

	Twelve paper rectangles, each 2 inches by 3+1/2 inches, 
	sufficient to make one Kepler's Star.

I was so amazed that the proportions for this project worked 
out within a sixteenth of an inch, that I wondered if 
variations of this folding technique would work. I found two 
more stars that make very nice decorations.

Several other types of stars (Projects VI. and VIII., as 
described below) can be constructed using variations of the 
techniques used to make Kepler's Star. Instructions for 
preparing the paper for these stars is fully described below, 
as an essential part of these additional projects.

Materials for all other projects are so basic, and no special 
tools are required. so instructions given for all of the other 
projects should be sufficient and complete, as described 

---- ----- ----- ----- ----- ----- ----- ----- ----- ----- 

I. MAKING 3-D SHAPES (Paper Sculpture) edit

Although this is the simplest activity, even graduate students 
at George Washington University found it extremely interesting 
when I shared it with them.

Cardboard shapes (carefully measured and cut out, each side 
about one inch long)
	Equilateral triangles
	Regular pentagons, hexagons, octagons
	Regular ten- and twelve-sided shapes 
	(Consult the source books to estimate
	the numbers required for each shape.) 
	Instructions and hints for making these are given above. 
Masking tape
Illustrations of the five regular and thirteen semi-regular 
(These may be compared with the three regular and eight 
	semiregular tessellations)

Scissors (to cut the masking tape)

Source books:
Ball, W. W. Rouse and H, S. M. Coxeter, _Mathematical_
	_Recreations_and_Essays_ (Thirteenth Edition), 
	Dover Publications, Inc., 1987, ISBN 
	0-486-25357-0 (pbk.) 
Fuse, Tomoko, _Multidimensional_Transformations_Unit_
	_Origami_, Japan Publications, Inc., 1990, 
	{{ISBN|0-87040-852-6}} (pbk.)
Wells, David, _The_Penguin_Dictionary_of_Curious_and_
	_Interesting_Geometry_, Penguin Books, ltd., 
	London, 1991, {{ISBN|0-14-011813-6}} (pbk.)


Begin Pictures illustrating the assembly of five polyhedra 


[2] CUBE




Ray Calvin Baker (talk) 22:42, 1 May 2012 (UTC)

End of Pictures illustrating the assembly of five polyhedra

Make the materials and tools available to the students. 
Construct a simple shape, such as a cuboctahedron, by 
sticking the necessary pieces together with squares of 
masking tape. During construction, show that 
parts of the structure can lie flat, until other parts 
are added, requiring that folds be made to allow the 
developing structure to take its final three-dimensional 
shape. Compare the final shape with its descriptive diagram.

Instruct the students to 
(1) select the shape they would like to build, 
(2) gather the necessary pieces, 
(3) cut squares of masking tape, and 
(4) assemble their model.

Inexperienced students may need to select additional pieces 
and cut additional masking tape. Accuracy in making the 
necesssary estimates comes with experience. Encourage 
cooperation: for example, one student may cut masking tape 
for several other students, with the understanding that he 
will receive help later, in building his own model. Note to 
helpers: Be a helper; don't "take over" someone else's model. 
Although neatness is commendable, any model which holds 
together and allows the student to see the relationships 
between the descriptive diagrams and the final, intended 
shape should be instructive. As time permits and interest 
persists, and supplies last, students may gain proficiency 
by building several models.


This is an open-ended activity, which COULD lead from the 
regular and semi-regular tessellations, and the regular and 
semi-regular polyhedra, prisms and anti-prisms, to the four 
Kepler-Poinsot polyhedra, to five more convex deltahedra, 
to 53 additional uniform polyhedra,  to 92 convex polyhedra 
with regular faces, not to mention compound polyhedra and 
other stellated polyhedra. There is a LOT of territory 
here, not all of it well known or thoroughly explored. 
And then, many of these can be used in the constuction 
of polytopes, of which there are 16 regular polytopes, etc. 
After all that, I'm sure I missed a few. And there are some 
of these shapes which I have never yet seen myself.

Some shapes may be rigid enough to leave some "windows" -- 
places where you deliberately do not tape in a shape. Instead, 
put a small knick-knack or an origami bird, flower, or angel
into your model for a different way to display your folding 

If this activity is going well, I may be able to demonstrate 
a few simple, traditional folds that create sequences of origami 
models, while some students are completing their paper 
sculptures. One example of this is "the multi-fold", which 
includes the oldest documented paper fold in Western culture, 
"Pajarita, the little Spanish bird". Another example is 
"the salt cellar" (formal title), which changes from 
"cootie catcher" to "the lover's knot", to "anvil", "sawhorse", 
and "crown". Another sequence, based on the "triple blintz fold", 
includes "perfume vial", "Japanese lantern", "Yokosan", and a 
"cross". Historically, such sequnces have inspired several 

---- ----- ----- ----- ----- ----- ----- ----- ----- ----- 


	(storigami: "Brothers Tall and Brothers Short")

Two sheets of 8+1/2 inch by 11 inch paper
	(One sheet for the box, one sheet for the lid)

Source book:
Sakoda, James Minoru, _Modern_Origami_, Simon and Schuster, 
	New York, NY, 1969, {{ISBN|0-671-20355-X}} (pbk.)

Since this is "storigami", the paper folding is intended to 
illustrate the story. The words of the story contain 
important clues concerning the sequence of folds, and the 
appearance of the paper after each fold (or series of folds).

	"Brothers Tall and Brothers Short"

(Stage directions -- instructions how to fold and display the 
paper -- are included between a pair of parenteses, like this. 
The actual story is enclosed in quotation marks. 
Give each student two sheets of ordinary 8+1/2 inch by 11 
inch paper. Invite them to watch carefully, and try to fold 
along, as the story is told. Try to pace the story, and 
intervene as necessary, so that no one gets left behind. 
This story is told in a way which will help everyone remember 
the essential steps. Adults and older children should find 
that the boxes with lids are extremely useful for storing 
household items, and items for hobbies and crafts.)

"This is the story of the Brothers Tall, 
Who didn't like extras creases at all."

(Place a single sheet of paper on the table in front of you, 
with the long edges running from left to right. 
Pick up the nearest edge, and place that edge exactly over 
the farthest edge. The paper should roll smoothly into a 
cylinder-like shape. Gently and carefully flatten the cylinder.
Make a single length-wise crease down the middle of the 
sheet of paper. This is a valley fold.

Note FYI: The first six creases, as described in the following 
steps, should all be valley folds, all facing upwards.

Lift your creased paper up off the table, and display the 
Brothers Tall. Let your imagination fill in the picture of 
the two brothers. Since this is the first crease, there are 
NO extra creases whatsoever, which the Brothers Tall dislike 
so much.)

"And they lived in a plain, long tent, 
To save money on rent."

(Display the plain, long tent shape formed by one crease. 
The tent shape clearly demonstrates that what is a valley 
fold on one side of the paper, is a mountain fold on the 
other side.)

"Each fell in love with a girl from next door; 
Soon they were married, now there are four."

(Make two more long creases to meet the previous crease 
in the middle. Note: When making the lid, leave a gap about 
the size of these words: "It is OK". Lift the paper to 
display the two couples. Let imaginations fill in the 
features of these two lovely couples.)

"They moved into a plain, long house, 
Because each had a spouse."

(Display the long house shape formed by the three parallel 

"When they went to the cupboard, the cupboard was bare.
There weren't even any shelves in there!"

(Hold the paper so the creases are all vertical. Open and 
close the cupboard doors. Notice that there are no shelves, 
because there are no extra creases.)

"That was the story of the Brothers Tall,
Who didn't like extra creases at all.
So short! So sad! Don't cry or make the paper wetter.
Just place it on the table, and give it a turn, 
I hope, a turn for the better."

(Place the paper flat on the table, then rotate it 90 degrees. 
This is the "turn for the better".)

"This is the story of the Brothers Short, 
Who liked to wear stripes just for sport."

(Make a single crease down the middle of the paper. This crease 
should cross the three creases left from the previous story. 
Let imaginations fill in the picture of the Brothers Short, 
but point out that the stripes are real -- the creases.)

"They lived in a short, striped tent,
To save money on rent."

(Display the short tent form, with its stripes.)

"Each fell in love with a girl from next door.
Soon they were married; now there are four."

(Make two more short creases to meet the short crease in the 
middle. When making the lid, leave a gap about the size of 
these words: "It is OK". Notice that the girls like stripes, 
too. What lovely couples!)

"They moved into a short, stiped house,
Because each had a spouse."

(Display the shape of the short, striped house.)

"They went to the cupboard; each shelf was filled with stuff.
Plenty of stuff, and plenty's enough."

(Hold the paper so that the three short creases are all 
vertical. Open and close the cupboard doors. The shelves 
are real (they are the creases left from the first part 
of the story), but you'll have to imagine the "stuff".)

"With enough in the cupboard, each family begins.
Soon each mommy is the mother of twins. 
Pick up each corner, and fold to the line,
You've done it just right, you've done it just fine.
Now pull up the blanket, over their toeses, 
Until all that sticks out is the tips of their noses. "

(Follow the instructions. My, what big noses these children 

"Turn everything 'round; the paper spins, 
so the other mommy can see both of HER twins.
Pick up each corner, and fold to the line,
You've done it just right, you've done it just fine.
Now pull up the blanket, over their toeses, 
Until all that sticks out is the tips of their noses."

(Follow the instructions. My, these children 
have big noses, too! Is it nice to tease? Of course not!)

"See how cleverly each corner locks.
Now, reach in and pull up, to open your box."

(Do I need to draw you a picture? This really is a very 
clever way to fold a box. To make a lid for your box, 
just take another sheet of paper, and repeat the story 
all over again, with two minor changes. Leave small gaps 
in the middle, "It is OK", when you make the folds which 
introduce the girls from next door. This will make the 
lid wider and longer than the box, but not quite as deep. 
Each lid has a folded rim, which can serve as a convenient 
label for the intended contents of each box. Just be careful 
to notice how the lid will fit on the box, so you don't 
write the label up-side-down!)

(Now that you have mastered this story, if you ever get paid 
for it, you will be a "professional boxer"! (A joke. Ha, ha.))

---- ----- ----- ----- ----- ----- ----- ----- ----- ----- 


	(other materials)

	75 Plastic soda straws (to make one ornament)
	white glue (optional)

	wire loop (narrow enough to fit through the hollow 
		straws. Instructions to make this tool are 
		given above.)
	Ball-point pen 
	Ruler (optional)

	Place two straws side-by-side. Estimate the location 
	of the center of the straws, then make a short mark 
	there with the ball-point pen. Turn one of the straws 
	end-for-end, then place it back beside the other 
	marked straw. If the marks line up, you did a good 
	job of finding the center. If the marks are off 
	by a fraction of an inch, estimate the location of the 
	place midway between the short marks, then make a 
	longer mark there. (This process should reduce your 
	error by half.) OR, use the ruler to measure off half 
	the length of the straws. Cut both straws in half at the 
	marked 	places. Use these half straws to measure off the 
	halfway point on thirteen more straws. Mark and cut 
	those straws in half.

	Now you should have 30 half length pieces, and 60 
	full length pieces.

	Push the wire loop down the hollow middle of one of the 
	short pieces. Thread one end of the string through the 
	wire loop, then pull the wire (with the string) back out 
	of the piece of plastic straw. Repeat this process two 
	more times, until you have three pieces threaded onto the 
	string. Tie a knot in the string, then pull it tight 
	(not TOO tight, or you may split a plastic straw!) so 
	that the three plastic pieces outline an equilateral 

	You should have something that looks like this. 
	(The dot indicates the location of the knot) 
	\  /

	Stretch out a length of string from the knot (about the 
	length of your arm should be fine.) You can tie on more 
	string any time, if you find that you need more string. 
	Just try to plan it so that your splices will be hidden 
	deep in the middle of a straw. Thread two more pieces 
	of straw onto the string. Tie another knot. Now you 
	should have something like this.
	\  /\

	Keep on threading short pieces of plastic straw and 
	tying knots until you have a network something like this.

	  c   c   c   c   c
	  /\  /\  /\  /\  /\        The letters mark 
	a/__\/__\/__\/__\/__\a      places where
	 \  /\  /\  /\  /\  /\      additional knots
	 b\/__\/__\/__\/__\/__\b    will be tied, as
	   \  /\  /\  /\  /\  /     described below.
	    \/  \/  \/  \/  \/
	     d   d   d   d   d

	Now take a short piece of string (about 4 inches long), 
	and loop it around the corners labelled "a". Pull it 
	tight, and tie a knot. This will pull the network into 
	something like a ring shape. Now take another short 
	piece of string and tie the corners labelled "b" 
	together. This should make the ring much easier to see. 
	Trim the ends of the strings, or tuck them down into 
	a nearby straw.

	At this time, you should have a nice ring-like structure, 
	with five loose flaps (marked "c") at the top, and five 
	more loose flaps (marked "d") at the bottom. Use a short 
	string to tie all five of the corners marked "c" together, 
	and use another short string to tie all five corners 
	marked "d" together. You should now have a neat, rigid, 
	little cage. This is the core of your ornament.

	The core has 20 equilateral triangles. Our next task is 
	to tie three full-length straws above the three corners 
	of each of these equilateral triangles. This should be 
	a fairly obvious matter of adding straws and tying knots. 
	I think the easiest to manage this in a systematic 
	fashion is this. Work on one triangle at a time. 
	(1) Take a piece of string about three times as long as 
	a straw. 
	(2) Tie one end of the piece of string to one of the 
	corners of the triangle you have elected to work on. 
	(3) Thread two full-length straws onto this string. 
	(4) Tie the loose end of the string to a second corner 
	of the triangle you are working on. 
	(5) Take another piece of string almost twice the 
	length of a straw. 
	(6) Tie one end to the third corner of the triangle 
	you are working on. 
	(7) Thread a full-length straw onto this string. 
	(8) Tie the loose end to the joint between the first 
	two straws you added in steps 2, 3 and 4. This should 
	position a rigid point above one of the 20 triangles 
	of the core of the ornament.

	Repeat this process for each of the remaining 19 

	You will probably wish to trim loose ends of string, or 
	tuck them out of sight. You may wish to leave a large 
	loop for hanging your ornament. A few drops of white 
	glue may help secure the knots and keep loose ends out 
	of the way.

	When you make your next star, see how many ways you can 
	figure out to save string, and make the tying of the 
	knots a more efficient process. I tried to keep things 
	as easy as possible for you, while we worked on your 
	first star.

---- ----- ----- ----- ----- ----- ----- ----- ----- ----- 


	(Compare with M. C. Escher's "Two Worlds".) 
	(Paper sculpture using business cards)

	12 cards or thick paper 2 inches by 3+1/2 inches 
		(Instructions for cutting these cards are 
		given above.)
	white glue or glue sticks (technically optional, 
		but highly recommended, especially for 





I may be the first to completely document this phase of the
construction. -- RCB

---- ----- ----- ----- ----- ----- ----- ----- ----- ----- 


        (quick, easy origami toy, with 
	a story: "The Lonely Little Japanese Lady")

3 inch by 5 inch index card

(Follow along with the story; adapted from the video, 
"A Peace of Paper".)

	"The Lonely Little Japanese Lady"

"Once upon a time there was a Japanese Lady, who would wake 
up every morning and begin her exercises. She lived by herself, 
and was lonely, so she was always hoping for company."

(Display a 3 by 5 index card, which represents the Japanese 

"She stretched out her right arm, then bent over and touched 
her left knee. Then she stood up straight again."

(Bend the top right corner of the card, so that what was the 
top edge lies over the left edge of the card. Crease, then 
unfold the card.)

"Then, she stretched out her left arm, bent over, and touched 
her right knee. Then she stood up straight again."

(Bend the top left corner of the card, so that what was the 
top edge lies over the right edge of the card. Crease, then 
unfold the card.)

""After many years of doing these stretching exercises every 
morning, she was so flexible that she could stretch out both 
arms, bend over backwards, and touch the backs of both knees. 
Then she stood up straight again."

(Don't you try this at home on yourself! You are not a three 
by five index card! But fold the card backward so that the 
corners, which were on top, lay on the creases left by the 
first two folds.)

"Would you like to see the little house the lonely Japanese Lady 
lives in? Just walk up to her door. There in the middle of the 
door, where the lines cross, is the doorbell button. Press the 
button to ring the doorbell, then pull the roof down into 
place to see the house."

(Hold the card up, with the top bent back slightly away from 
you. This represents the door, with lines that cross in the 
middle. Press the doorbell button. The two sides of the card 
should snap toward you. Pull the roof down into position, and 
look at the little house.)

"Now, imagine that the little Japanese Lady has come to the 
door. She feels that it is rather chilly outside today, so she 
wraps her shawl about herself. 'Would you like to come inside 
to warm up?', she asks. 'I'm sorry. I have other things I must 
do today, so I must be on my way. Perhaps you will have another 
visitor today.', you reply."

(Pull both vertical edges of the "house" forward, so that these 
edges of the card meet in the middle. Crease the new folds 

"The little Japanese lady still feels chilly, so she claps her 
hands together in front of her, several times."

(Pull the two triangular flaps, which represent her arms, 
forward several times, as if she's clapping her hands.)

"Thinking she is a bit stiff from the chill, she repeats one of 
her stretching exercises. She bends backward so far that the 
top of her pointed hat touches the bottom of her heels."

(Bend the pointed top of the card back until it touches the 
bottom of the card, in the middle.)

"Thinking that more vigorous excercise may help her warm up, 
she leaps into the air, and kicks out both her feet so far 
that her toes touch her tummy. She lands quickly and 
gracefully on her feet."

(Bend the bottom of the folded card forward, so that the lowest 
edge meets the crease which marks the lady's waist. Fold the 
card into a compact shape, to suggest how she lands quickly 
and gracefully.)

"When the Japanese Lady turns around again, she sees that, 
indeed, she does have another visitor today. There, on the 
doorstep, she sees a little bunny. She leans down to pet the 
bunny, but it hops away."

(In its compact shape, the folded card resembles a rabbit, 
with big ears. If you stroke your finger down its back, the 
bunny may hop for you!)

"'Perhaps I will see him again tomorrow', the Lady says to 
herself. Indeed, she will, if you take another three by five 
index card, and share this story with someone tomorrow."

---- ----- ----- ----- ----- ----- ----- ----- ----- ----- 


	(a starry paper sculpture 
	from ordinary sheets of paper)


---- ----- ----- ----- ----- ----- ----- ----- ----- ----- 



	(I may be among the first to completely document 
	this important phase of the construction.)

[ INCOMPLETE! but I have an old file, in which I began to 
describe this project. -- RCB ]

---- ----- ----- ----- ----- ----- ----- ----- ----- ----- 


	(Compare with M. C. Escher's prints, "Gravitation" 
	and "Order and Chaos".)
	(The beauty of modular origami is that the same 
	module can be assembled in several, completely 
	different ways. Learn to fold just one module; 
	but be able to learn how to make several 
	different models using that module.)


---- ----- ----- ----- ----- ----- ----- ----- ----- ----- 

IX. A SIMPLE JUMPING FROG (origami toy) edit


---- ----- ----- ----- ----- ----- ----- ----- ----- ----- 

X. ANGEL (a simplified origami ornament or finger puppet) edit


---- ----- ----- ----- ----- ----- ----- ----- ----- ----- 


	(authentic traditional Japanese origami)


The end.

Ray Calvin Baker (talk) 20:45, 13 April 2012 (UTC)