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Twirl Kusudamaa tool for polyhedra understanding
Krystyna & Wojciech [email protected]
www.origami.edu.plPolish Origami Association
5th International Convention for DIDACTICS OF PAPERFOLDING FOR
EDUCATORS
Freiburg im Breisgau, GermanyNovember 12th -14th, 2010
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Contents
1. Folding a model. An example of twirl kusudama
2. Relation between an origami model and a mathematical
object.
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Let’s fold a model
Simple Twirl Kusudama
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A module
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1 2 3 4
5 6
7 8
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A macro-module
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Assembly – tetrahedron structure
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It’s time for structure
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Geometrical structure of the model
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Macro-modules as polygons
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Macro-modules as polygons
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Macro-modules as polygons
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Macro-modules as polygons
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Macro-modules correspond to polygons
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A Polygon or a Flower?
Macro-modules: triangle (6 modules) and hexagon (12 modules)
Kusudama 4x6 with holes from 48 modules with petals
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A flower or a polygon?
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A flower or a polygon?
What kind of polygons is it?
What kind of polyhedrons is it?
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What polyhedron is it?
A cube
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What polyhedron is it?
An octahedron
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What polyhedron is it?
A cuboctahedron
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What polyhedron is it?
A truncated cube
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Different interpretations
There is no unique proper interpretation of an origami model in
geometric terms.A school problem is usually artificial. It is
simplified and there is only one proper answer.Twirls offers an
environment that is still simple and accessible in math teaching
but closer to real life problems where usually there is more than
one proper answer.
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A Polygon or a Flower?
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A Polygon or a Flower?
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A Polygon or a Flower?
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A Flower or a Polygon?
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A Polygon or a Flower?
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A Polygon or a Flower?
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A vortex as a polygon
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A vortex as a polygon
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A vortex as a polygon
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The shape of a face – a polyhedral structure
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The shape of a face – a polyhedral structure
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Face modulesVertex modulesEdge modules
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A face or o vertex?
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A face or o vertex?
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Edges, faces or flowers?
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Different length of spirals and polyhedral structure
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Struktury wielo ścienne a moduły o różnej długo ści spiral
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Can you see structure ?
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Can you see
a polyhedron?
Kusudama 8x6
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Can you see
a polyhedron?
Kusudama 8 x 6 triangles + 6 squares
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Can you see
a polyhedron?
Kusudama from 210 modules
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Can you see
a polyhedron?
Kusudama 18 squares
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Can you see
a polyhedron?
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Can you see
a polyhedron?
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Can you see
a polyhedron?
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Can you see
a polyhedron?
Kusudama from 210 modules
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Can you see
a polyhedron?
Geodesics from edge modules
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Can you see
a polyhedron?
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Can you see
a polyhedron?
Duality
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Can you see a
polyhedron?
Similar appearance, but different structure
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Can you see
a polyhedron?
Different appearance, but the same structure
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Polyhedra properties discovered through models with spirals
• We understand polyhedra better both through folding models and
through looking at them.
• When we build models we have to use polyhedra structures.
• Regular polygon is inscribed into circle, so spirals used for
a polygon (face of a polyhedron) should be the same lenght.
• Regular polyhedra are inscribed into a sphere (Geodesics are
the best example). Twirl polyhedra models form floral balls
(kusudama = floral ball).
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Twirls in math education
Twirl kusudamas are visual appealing models, what enhance
student motivationGeometric structure is essential to get a model.
Such structure may be hidden in the educational process behind a
visual goal of origami model, but students learn about polyhedra
structures as away to make something different (in their mind) from
geometry.Origami model may be interpreted in geometrical terms in
different ways. There is no unique proper answer, what is common to
the real life problems but very rare in case of mathematical
problems used in mathematics’ teaching.
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More about twirl kusudama
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Thank You!
