· Activity
Name: Deltahedron Investigation
· Objectives:
Students
will discover the different Platonic solids built with equilateral triangles as
well as several other models. They will
also discover the pattern for determining the number of edges given the vertices
and faces. Students will become more
aware of solids and learn vocabulary associated with them.
· EALR/Standards:
· Materials:
Triangle
Polydrons (30 per group)
Chart
Page
· Teacher
Notes
o
Prerequisites for the learner:
Basic definitions for edge, vertex, face,
concave, and planar.
o
Teacher hints for the activity:
Allow extra time at the start
for exploring with the polydrons. To
test for aplanar faces, set the model on a flat surface and see if it “rocks”
between each set of triangles. There is
one model where two triangles are just barely aplanar and the rock test will
verify this better than just eyeballing.
o
Introductory questions:
Name some solids that you have seen or worked with.
What kinds of shapes make up the sides
and bottoms of solids?
If we are limited to triangles, how many solids do
you think we can make?
If we limit ourselves to convex solids
made with one triangle per face, how many solids do you think we can make?
o
Wrap-up questions:
What patterns can you see in the chart
page?
Why could we not make one of the models
that our pattern says exists?
Where have you seen solids made of
triangles?
o
Solutions:
See worksheet page four.
o
Assessment suggestions:
As
a class answer the first worksheet chart page, and then use the second and
third pages to check for student understanding, thinking, and analysis.
· The
Activity:
Students
begin this activity by getting the polydrons and chart papers. If the polydrons are new to the students,
allow a few minutes for explorations. Begin
by discussing that a deltahedron is any polyhedron made solely of equilateral
triangles. For this project, the
polydrons will also need to be aplanar (No faces which share edges are planar)
and convex. Have the students work in
their groups on the building of different models and recording the
information. If a group says they are
done, ask them to put their data in some type of order and try to numerically
prove that they have indeed found all of the possibilities.
The
factoid sheet can provide students with clues about the different models. Ask the students to find at least one model
that fits each description. Each model
may be used several times, and more than one model may fit each
description.
When
a group is ready for extensions, have them work on the second page of the chart
answering the extension questions below.
When the class begins to
quit, gather the data on the board and work with the students on the
pattern.
· Assessment
material:
· Extensions:
What
happens if squares are used?
Pentagons? Mixtures of squares
and triangles? Pentagons and triangles? Others?
Deltahedron Investigation
Definitions:
Deltahedron:
Any Polyhedron in which every face is an equilateral triangle.
Aplanar: Any polyhedron in which two adjoining faces
are not planar.
Convex: Every face will lie flat on a tabletop and
will roll to each adjoining
face.
Investigate and
record data on as many convex, aplanar
deltahedra as you can find. If you
run out of ideas, try arranging the data in some type of order and see if there
are any holes in your data.
Vi = Number of
i-valent vertices. (I.e. V3 = Vertices with 3 faces meeting)
V = Total number
of vertices. E = Number of edges. F = Number of faces.
Deltahedron Page Two
After looking
over your data list, are you able to find any patterns? If so, describe what
you have found
._________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
Were you able to
build all of the models that your data claims exist? Why or why not?
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
Is there a
reason that you can see that explains why you have found all possible
models?
What is
it?______________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
What would
happen if you only used squares?
_________________________________
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
What would
happen if you only used pentagons? ________________________________
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
Build a couple
of the models again, measure the dihedral angles of your models. Are
there any
connections? Are there shapes that have
only one dihedral angle measurement?
How about
exactly two?____________________________________________________
_______________________________________________________________________
_______________________________________________________________________
Which of the
shapes that you made would be considered regular shapes? Why?________
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
Deltahedron Factoids
The following
statements apply to one or more of the deltahedra.
1. I look like two pentagonal pyramids stuck
together. ______________________
2.
I am a
4-gonal dipyramid. ______________________
3.
I look like
a square-based pyramid stuck on each square face of a triangular prism.
______________________
4.
One of my
cross-sections is a regular hexagon. ______________________
5.
I am a
3-gonal antiprism. ______________________
6.
I look like
a square-based pyramid stuck on each base of a square antiprism.
______________________
7.
I look like
a pentagonal-based pyramid stuck on each base of a pentagonal antiprism.
______________________
8.
I look like
triangular pyraminds stuck on the bases of a triangular antiprism.
_____________________
9.
Each of my
vertices has the same odd valence. ______________________
10.
I am a
regular polyhedron. ______________________
11. I have twelve faces. My four 5-valent vertices are connected to
each other around the equator.
_____________________
12. I am the dual of a regular polyhedron. ______________________
13. One of my cross sections is identical to
each of my faces. ______________________
14. I would make a fair die, but I am not a
regular polyhedron. _____________________
15. I have an even number of faces, but no two
of my faces are parallel. _____________
16. If you cut me in half in the right way you
would have two congruent copies of another deltahedron.
______________________
Deltahedron Investigation
Definitions:
Deltahedron:
Any Polyhedron in which every face is an equilateral triangle.
Aplanar: Any polyhedron in which two adjoining faces
are not planar.
Convex: Every face will lie flat on a tabletop and
will roll to each adjoining
face.
Investigate and
record data on as many convex, aplanar
deltahedra as you can find. If you
run out of ideas, try arranging the data in some type of order and see if there
are any holes in your data.
Vi = Number of
i-valent vertices. (I.e. V3 = Vertices with 3 faces meeting)
V = Total number
of vertices. E = Number of edges. F = Number of faces.
4 0 0 4 6 4
Tetrahedron
2 3 0 5 9 6
Triangular
bipyramid
0 6 0 6 12 8
Octahedron
0 5 2 7 15 10
Pentagonal
bipyramid
0 4 4 8 18 12
Snub disphenoid
0 3 6 9 21 14
Tri Augmented
Triangular prism
0 2 8 10 24 16
Gyro elongated
square bipyramid
0 1 10 11 27 18
* Not possible
0 0 12 12 30 20
Icosahedron
Deltahedron Factoids
The following
statements apply to one or more of the deltahedra.
1. I look like two pentagonal pyramids stuck
together. __Pentagonal bipyramid__
2. I am a 4-gonal dipyramid. ___Octahedron_________
3. I look like a square-based pyramid stuck on
each square face of a triangular prism.
_______Tri Augmented Triangular Prism_
4. One of my cross-sections is a regular
hexagon. ______Octahedron________________
5. I am a 3-gonal antiprism. ____Octahedron__________________
6. I look like a square-based pyramid stuck on
each base of a square antiprism.
_______Gyro elongated square
bipyramid______
7. I look like a pentagonal-based pyramid stuck
on each base of a pentagonal antiprism.
____Icosahedron__________________
8. I look like triangular pyraminds stuck on
the bases of a triangular antiprism. __Snub disphenoid_
9. Each of my vertices has the same odd
valence.___Tetrahedron, Triangular bipyramid, Icosahedron_
10. I am a regular polyhedron. ___Tetrahedron,
Octahedron, Icosahedron______________
11. I have twelve faces. My four 5-valent vertices are connected to
each other around the equator.
____Snub disphenoid__________________
12. I am the dual of a regular polyhedron. ___Tetrahedron, Octahedron,
Icosahedron_________
13. One of my cross sections is identical to
each of my faces._Triangular bipyramid, Snub disphenoid__
14. I would make a fair die, but I am not a
regular polyhedron.
_______Triangular bipyramid,
Pentagonal bipyramid_______
15. I have an even number of faces, but no two
of my faces are parallel.
_Tetrahedron, Triangular bipyramid,
Pentagonal bipyramid, Gyro elongated square bipyramid__
16. If you cut me in half in the right way you
would have two congruent copies of another deltahedron.
__Triangular
bipyramid___
