Origami/Examples 1
< Origami
FLYER
[edit | edit source]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! THIS IS NOT A COMPETITION! It's a CO-OPERATION! 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.
PLAN OF EVENTS
[edit | edit source]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. ---- ----- ----- ----- ----- ----- ----- ----- ----- -----
THE ORIGAMI TOOL KIT
[edit | edit source]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) Scissors Toothpicks Skewers Tweezers 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.) ---- ----- ----- ----- ----- ----- ----- ----- ----- ----- NOTES TO PARENTS AND TEACHERS
HOW TO PREPARE MATERIALS
[edit | edit source]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 use. I am planning to bring prepared materials for these special projects, in order to allow you to begin working without delay.
I. SUMMARY OF POPULAR 3-D SHAPES
[edit | edit source]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 Cuboctahedron 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 icosadodecahedron 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. </PRE. == MAKING POLYGONS FOR 3-D SHAPES == <PRE> Materials: Ordinary Poster Board (white, or colored, one or both sides. What I get in Mayland, USA, is usually 22 inches by 28 inches.) Tools: 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 Scissors Procedure: GENERAL HINTS 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 substitute.
WONDERFUL SUBSTITUTE
[edit | edit source] 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
airplanes. START YOUR OWN TOY FACTORY!
ADDITIONAL BENEFITS:
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
techniques.
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
colors.
Materials:
Poster board (Do NOT try to use foam-core board for
these projects!)
Tools:
Yard stick
Ball-point pen, or pencil
Scissors
TO MAKE SQUARES
[edit | edit source]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). Procedure: (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. RIGHT WAY WRONG WAY +------------+ +------------+ 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 drawn. (5) Cut each strip into squares. Yield: A piece of poster board 22 inches by 28 inches should make 616 1-inch squares.
TO MAKE TRIANGLES AND HEXAGONS
[edit | edit source] 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
. ________________________________.
/ \ / \ / \ / \ /
/ \/ \/ \/ \/
/\ /\ /\ /\ /
/__\____/__\____/__\____/__\____/
/ \ / \ / \ / \ /
/ \/ \/ \/ \/
/\ /\ /\ /\ /\
/__\____/__\____/__\____/__\____/__\
/ \ / \ / \ / \ / \
/ \/ \/ \/ \/ \
/\ /\ /\ /\ /\ /\
c/__\____/__\____/__\____/__\____/__\____/__\.e
Procedure:
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)
long.
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.
Yield:
I got [ ?? -- RCB ] hexagons and [ ?? -- RCB ] equilateral
triangles from my sheet of poster board. There
was some scrap near the edges of the sheet.
TO MAKE REGULAR PENTAGONS
[edit | edit source] 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.
C
g+------_+-----+j
| _- |\ | 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.
A+_------|--/--+
| -_ | / |
| -_|/ | 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
measurements).
C
g+------_+-----+j
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
+-------+-----+
E
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)
g+-----------_+C
H * |90 18_-
sin | _-
(18) |72_- H
B+-
(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)
C+-------+j
\ |
\ | H *
\ | cos
H \ | (18)
\36|
\ |
\|
+D
(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.
SOURCE CODE FOR THE QB64 BASIC COMPILER:
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
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
PRINT "25.4 * cos(36); 25.4 * sin(36):"
PRINT 25.4 * COS(36 * Pi / 180)
PRINT 25.4 * SIN(36 * Pi / 180)
PRINT
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.)"
END
RESULTS OF RUNNING THE PROGRAM:
Pi = 3.141593
This program calculates dimensions for grid to make regular pentagons.
16 * cos(36); 16 * sin(36):
12.94427
9.404564
16 * cos(18); 16 * sin(18):
15.2169
4.944272
(Dimensions in sixteenths of an inch.)
25.4 * cos(36); 25.4 * sin(36):
20.54903
14.92975
25.4 * cos(18); 25.4 * sin(18):
24.15684
7.849032
(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!
[ INSERT PICTURE "STAR.jpg" HERE. -- RCB]
Here is how to find the necessary dimensions
WITHOUT ARITHMETIC!
(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
paper.
(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
went).
(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.
Procedure:
(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.
Yield:
I got [ ?? -- RCB ] regular pentagons from my sheet of
poster board. Some scrap had to be trimmed from each
rectangle.
Ray Calvin Baker (talk) 03:14, 15 May 2012 (UTC)
TO MAKE REGULAR OCTAGONS
[edit | edit source] +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)
Procedure:
(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.
Yield:
Four triangles of scrap had to be trimmed from each square
to make [ ?? -- RCB ] regular octagons -- "stop signs".
TO MAKE REGULAR 10-SIDED POLYGONS
[edit | edit source]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: Procedure: Yield: [ INCOMPLETE! -- RCB ]
TO MAKE REGULAR 12-SIDED POLYGONS
[edit | edit source]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: Procedure: Yield: [ INCOMPLETE! -- RCB ]
III. MAKING TOOLS FOR BUILDING A TRADITIONAL FOLK ORNAMENT
[edit | edit source]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. Materials: Wire (light gauge doorbell hookup wire from a hardware store works just fine) Tools: Wire cutters Pliers (needle-nosed pliers work best) Ruler Ball-point pen or pencil Procedure: (1) Measure off a piece of bell wire about 50 per cent longer than a drinking straw. (Drat! More of that arithmetic!) (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.
IV. PREPARING MATERIALS TO MAKE KEPLER'S STAR
[edit | edit source]Materials: Take one sheet of paper 8+1/2 inches by 11 inches for each star you wish to make. Tools: Ruler Ball point pen or pencil Scissors Procedure: 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. Yield: 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 below. ---- ----- ----- ----- ----- ----- ----- ----- ----- -----
I. MAKING 3-D SHAPES (Paper Sculpture)
[edit | edit source]Although this is the simplest activity, even graduate students
at George Washington University found it extremely interesting
when I shared it with them.
Materials:
Cardboard shapes (carefully measured and cut out, each side
about one inch long)
Equilateral triangles
Squares
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
polyhedra
(These may be compared with the three regular and eight
semiregular tessellations)
Tools:
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.)
Procedure:
Begin Pictures illustrating the assembly of five polyhedra
[1] TETRAHEDRON
![[1]](https://commons.wikimedia.org/wiki/File:Platonic_Solids_Stereo_1_-_Tetrahedron.gif){kind=link}
[2] CUBE
![[2]](https://upload.wikimedia.org/wikipedia/commons/8/81/Platonic_Solids_Stereo_2_-_Cube.gif){kind=link}
[3] OCTAHEDRON
![[3]](https://upload.wikimedia.org/wikipedia/commons/0/0d/Platonic_Solids_Stereo_3_-_Octahedron.gif){kind=link}
[4] TWELVE REGULAR PENTAGONS
![[4]](https://upload.wikimedia.org/wikipedia/commons/5/52/Platonic_Solids_Stereo_4_-_Dodecahedron.gif){kind=link}
[5] TWENTY EQUILATERAL TRIANGLES
![[5]](https://upload.wikimedia.org/wikipedia/commons/4/44/Platonic_Solids_Stereo_5_-_Icosahedron.gif){kind=link}
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.
SHAPES IN SPACES
[edit | edit source]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 skills. 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 story-tellers. ---- ----- ----- ----- ----- ----- ----- ----- ----- -----
II. A MOST USEFUL ORIGAMI MODEL
[edit | edit source] A BOX WITH LID
(storigami: "Brothers Tall and Brothers Short")
Materials:
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.)
Procedure:
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
creases.)
"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
have!)
"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.))
---- ----- ----- ----- ----- ----- ----- ----- ----- -----
III. A TRADITIONAL FOLK ORNAMENT
[edit | edit source]THIRD STELLATION OF THE REGULAR PENTAGONAL DODECAHEDRON (other materials) Materials: 75 Plastic soda straws (to make one ornament) String white glue (optional) Tools: Scissors wire loop (narrow enough to fit through the hollow straws. Instructions to make this tool are given above.) Ball-point pen Ruler (optional) Procedure: CUT 15 STRAWS IN HALF 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. BUILD THE CENTRAL CORE OF THE ORNAMENT 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 triangle. 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. ADD THE STAR-LIKE POINTS 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 triangles. FINAL FINISHING 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. ---- ----- ----- ----- ----- ----- ----- ----- ----- -----
IV. KEPLER'S STAR
[edit | edit source](Compare with M. C. Escher's "Two Worlds".) (Paper sculpture using business cards) Materials: 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 beginners.) Procedure: FOLD EACH CARD [ INCOMPLETE! -- RCB ] AsSEMBLE THE FOLDED CARDS [ INCOMPLETE! -- RCB ] I may be the first to completely document this phase of the construction. -- RCB ---- ----- ----- ----- ----- ----- ----- ----- ----- -----
V. THE HOPPING BUNNY
[edit | edit source](quick, easy origami toy, with a story: "The Lonely Little Japanese Lady") Materials: 3 inch by 5 inch index card Procedure: (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 lady.) "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 firmly.) "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." ---- ----- ----- ----- ----- ----- ----- ----- ----- -----
VI. A STAR REVISITED
[edit | edit source](a starry paper sculpture from ordinary sheets of paper) [ INCOMPLETE! -- RCB ] ---- ----- ----- ----- ----- ----- ----- ----- ----- -----
VII. AN INTRODUCTION TO MODULAR ORIGAMI
[edit | edit source]Part One: A SIMPLE MODULE Part Two: ASSEMBLING SEVERAL MODULES (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 ] ---- ----- ----- ----- ----- ----- ----- ----- ----- -----
VIII. THE FIRST STELLATION OF THE REGULAR PENTAGONAL DODECAHEDRON
[edit | edit source](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.) [ INCOMPLETE! -- RCB ] ---- ----- ----- ----- ----- ----- ----- ----- ----- -----
IX. A SIMPLE JUMPING FROG (origami toy)
[edit | edit source][ INCOMPLETE! -- RCB ] ---- ----- ----- ----- ----- ----- ----- ----- ----- -----
X. ANGEL (a simplified origami ornament or finger puppet)
[edit | edit source][ INCOMPLETE! -- RCB ] ---- ----- ----- ----- ----- ----- ----- ----- ----- -----
XI. THE SITTING CRANE
[edit | edit source](authentic traditional Japanese origami) [ INCOMPLETE! -- RCB ] The end.
Ray Calvin Baker (talk) 20:45, 13 April 2012 (UTC)