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Sonobe Origami for enriching understanding of geometric
concepts in three dimensions
DONNA A. DIETZ
American University
Washington, D.C.
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Donna Dietz, American UniversitySonobe Origami for enriching understanding of geometric concepts in three dimensions
St. Mary's College of MarylandNovember 7, 2015. MAA Section Meeting3:15 pm -3:35 pm Goodpaster Hall Room 109
"Programs that take advantage of paper folding to teach mathematics are thriving in many parts of the world," according to the organizers of the MAA origami-themed Contributed Paper Session to be held at the JMM in January 2016. But, K-12 should not be having all the fun! In this talk, I will show some ways of stimulating student engagement using sonobe origami. These activities can be used in General Education mathematics "appreciation" courses or for Non-Euclidean Geometry for mathematics majors. Specifically, the goals are enriching student understanding of surface curvatures and helping them understand the duality of the regular polyhedra using these folded paper objects.
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Things to chat about...
● Construction overview and classroom hints
● Duality of regular polyhedra
(aka Platonic Solids)
● Euler's χ for topological classification
● Angle deficit for topological classification
● Gauss-Bonnet Theorem
● Bonus Puzzle!
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Construction Basics
● tinyurl.com/sonobe
● I also have videos on YouTube
● for Gen Ed course- one class period
● students spend about an hour outside of class
● 30 pieces of paper per student
● chirality compatibility is a construction issue
● assembly is a learning experience for students
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The Paper I Use in Class:
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Pros and Cons
● No glue
● No scissors
● Straightedge and Compass is equivalent to folding
● Pride in crafting
● Only time for one large project rather than many small ones
● Object is more complex than regular polyhedra
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Make 30 of those
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Ok, Great!
Now, what is thissonobe project
good for?
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Duality of Regular Polyhedra
mathforum.org
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Cube with Octahedron
behance.net
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Icosahedron with Dodecahedron
behance.net
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Tetrahedron is self dual
math.brown.edu
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mathcraft.wonderhowto.com
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How much paper is that!?
To build the basic object, consider the underlying icosahedron (20 sides) and count pyramids.
Each side has a pyramid attached which uses 3 units. But each unit is used in two pyramids.
20*3/2 = 30
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Other objects
● Tetrahedron: 4*3/2 = 6 units, but it's a cube!
● Octahedron: 8*3/2 = 12 units
● Icosahedron: 20*3/2 = 30 units
● Massive ball: (12x5)*3/2 = 90 units
All of these objects are based off of an underlying triangular mesh frame.
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Euler's χ for topological classification
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Euler's χ for topological classification
● 6 vertices = v
● 9 edges = e
● 4 faces = f
● χ = v – e + f
● χ = 1
for single components
with no holes
maverick.inria.fr
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Euler's χ for topological classification
● 16 vertices = v
● 28 edges = e
● 14 faces = f
● 2 “blobs” = b
● χ = v – e + f - b
● χ = (16+14)–(28+2) = 0
for “donuts”
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Euler's χ for topological classification
● 8 vertices = v
● 12 edges = e
● 6 faces = f
● 1 “blob” = b
● χ = v – e + f - b
● χ = 8 – 12 + 6 – 1 = 1
for “spheres”
mathcraft.wonderhowto.com
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Euler's χ for topological classification
● 6+8 vertices = v
● (3+3/2)*8 edges = e
● 8*3 faces = f
● 1 “blob” = b
● χ = v – e + f - b
● χ = 14 - 36 + 24 – 1 = 1
for “spheres”
mathcraft.wonderhowto.com
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Euler's χ for topological classification
● 20 + 12 vertices = v
● (3+3/2)*20 edges = e
● 20*3 faces = f
● 1 “blob” = b
● χ = v – e + f - b
● χ = 32 - 90 + 60 – 1 = 1
for “spheres”
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Angle deficit for topological classification
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Angle deficit for topological classification
“And now for somethingcompletelydifferent...”
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Angle deficit for topological classification
● Imagine a flattened cube
● At each vertex, 90 degrees is “missing”
● 8 * 90 = 720
● Total Angle Deficit is
720mathcraft.wonderhowto.com
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Angle deficit for topological classification
● A flattened cube!
● At each vertex, 90 degrees is “missing”
● 8 * 90 = 720
● Total Angle Deficit is
720
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Cuboctahedron Template
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Cuboctahedron in 3D
● 12 vertices
● 60 degree deficit per
● 12*60=720
● Total Angle Deficit is
720
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Sonobe!
● 20 vertices with
90 degree deficit
(elliptic curvature!)
● 12 vertices with
-90 degree deficit
(hyperbolic curvature)
● 20*90-12*90=720
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But....
Total Angle Deficit = 4π * χ
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Gauss-Bonnet Theorem
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Gauss-Bonnet Theorem
If we trace out a closed curve on a surface, the total enclosed Gauss curvature (total angle deficit) is 2π minus the total angle defect (angles of deflection) around the curve.
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Gauss-Bonnet simple example
● Total angle defect is zero. (This paper ribbon is a geodesic.)
● Each half of the sphere must have the same enclosed Gauss curvature.
● Total must be 4π
● Each half contains
2π
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Gauss-Bonnet simple example
● Total angle defect is zero. (This paper ribbon is a geodesic.)
● Each half of the can must have the same enclosed Gauss curvature.
● Total must be 4π
● Each half contains
2π
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Gauss-Bonnet simple example
● Total angle defect is zero.
● Each half of the cube must have the same enclosed Gauss curvature.
● Each half contains 2π
● Each vertex has π/2
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Gauss-Bonnet example
● Total angle defect is:
2(180-72)+2(180-120)
336
● 360-336=24 degrees
● Angle deficit:
360-(60+60+108+108)
24 degrees
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Gauss-Bonnet example
● Total angle defect is:
3(180-84)
288
● 360-288=72 degrees
● Angle deficit:
3(360-60-60-108-108)
720 degrees
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Gauss-Bonnet example
● Total angle defect is:
5(180-120)
300
● 360-300 = 60 degrees
● Angle deficit:
5(360-108-120-120)
60 degrees
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Gauss-Bonnet example
● Total angle defect is zero (geodesic!)
● 360 degrees is promised
● Angle deficit:
5(90)+(-90)
360 degrees
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Bonus Puzzle
Use 3 colors, 10 pieces of each color.
Try to make sure you get exactlyone of each coloron each pyramid.
(Solve using a graph first.)
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Bonus Puzzle Solution
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http://www.donnadietz.com
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