Unit on Euler Characteristic and Polyhedra
Do these on or about March 31
Polyhedra and Euler Characteristic
Task 1--Start this before the
discussion session: Get out the Polydrons
you took home with you,
and build some polyhedra. Count the vertices (V), edges (E) and
faces (F) of your polyhedra and enter them in this worksheet. Save the
worksheet when you're done.
📜Assignment 1: Turn in the worksheet
by uploading
it to the assignment Euler-Characteristic-1
in Canvas.
Optional: Video discussion
online at 2:00 March 31st. It will be recorded and available in
Canvas afterwards. Get out your Polydrons for this discussion!
Nets of Polyhedra
Task 2: Learning about and
visualizing nets:
If you take a polyhedron that you built out of polydrons, and
un-snapped it part way, so that it was still all in one connected
piece, but it lays flat on the table, that would be a net for that
polyedron
🎮
Practice visualizing flat nets and their corresponding 3D polyhedra by
playing Nets of 3D Shapes at TurtleDiary.com:
https://www.turtlediary.com/game/nets-of-3d-shapes.html
(nothing to turn in)
Task 3: Designing nets in
Geogebra
🎥
First watch this video of me making
the net of a dodecahedron using Geogebra.
Then go to https://www.geogebra.org/
and make nets of each of these three polyhedra:
Copy each of your nets into a Word document and save it
📜Assignment 2: Turn in the three
nets you designed
by uploading the document in the Canvas assignment Polyhedra-Nets
Do these on or about April 2
Optional: Video discussion
online at 2:00 April 2nd. It will be recorded and available in
Canvas afterwards.
Join Zoom Meeting
https://zoom.us/j/999092158
Meeting ID: 999 092 158
Euler Characteristic of Planar Graphs
🎥
Watch Video 1: An example of drawing an
appropriate graph, and calculating the Euler Characteristic.
∴ Assignment 3: Do the Making
Connected Planar Graphs task in our Geogebra group
If you have not yet joined the Geogebra group, go to: https://www.geogebra.org/groups and enter the code DHHRB
🎥
Watch Video 2:
the most common way of
proving that the Euler characteristic of a connected planar graph is 2.
🎥
Video 3: Watch this video made by 3Blue1Brown on Euler's Formula and
Graph Duality
📜
Assignment 4: Fill in
this worksheet that asks you
to work
through some of the details in the proofs in videos 2 and 3.
Save your completed worksheet and Turn
in the worksheet by uploading it to the Canvas assignment Euler-Characteristic-Proofs
Connecting Planes, Spheres and
Polyhedra
🎥 Video 1: The
vertices, edges and faces of a polyhedron
can be drawn onto/projected onto a sphere.
The number of vertices, edges and faces of the polyhedron will be the
same as the number of vertices, edges and faces of the graph on the
sphere. The Euler characteristic of a polyhedron is the same as
the Euler Characteristic of a sphere.
🎥
Video 2: You can project things on a sphere onto the
plane. This is a stereographic projection.: https://www.youtube.com/watch?v=VX-0Laeczgk
🎥
Video 3: Further adventures in stereographic projection (you can
probably just let the previous video go to the "next" video to get to
this one: https://www.youtube.com/watch?v=lbUOScpu0ws
🎥
Video 4: Using stereographic projections, you can turn a graph in the plane into a graph on the
sphere.
- You can also turn a graph on a sphere into a graph in the
plane. These graphs will always have the same number of vertices,
edges and faces, and the "outside" face of a planar graph makes more
sense when you look at it on a sphere.
- Because we know (proved) that the Euler characteristic of a
planar graph is always 2, then we know that the Euler characteristic of
a (connected, embedded) graph on a sphere is always 2. We say the Euler
characteristic of a sphere is 2.
- Then if we have a polyhedron where you can project its vertices,
edges and faces onto a sphere, its Euler characteristic is the same of
the Euler characteristic of the sphere (which is the same as the Euler
characteristic of a graph in the plane) which is 2!.
- This is so cool and useful, that we implicitly add this into the
definition of a polyhedron, so a polyhedron isn't just something that
has flat faces and straight edges, and is enclosed, but it's also
something that can be projected onto a sphere, so its Euler
characteristic is 2.
Do these on or around April 7
Optional: Video discussion
online at 2:00 April 7th. It will be recorded and available in
Canvas afterwards.
Join Zoom Meeting
https://zoom.us/j/716087176
Meeting ID: 716 087 176
Shapes that have an Euler characteristic ≠ 2
∴ Start by going to the
Geogebra group and doing the Experiment
with donuts task.
Then watch these videos about finding the Euler Characteristic of a
torus and a 2-holed
torus:
🎥
Video 1: Another torus with flat faces
🎥
Video 2: An efficient way to cut up a round
torus
🎥
Video 3: One way of cutting up a 2-holed
torus
Check out Wikipedia for some pictures of surfaces with
genus 0, 1, 2 and 3
📜
Pencil and paper assignment:
- Find another way of cutting up a torus into vertices, edges and
faces. Show your way, and use it to show that the Euler
Characteristic of a torus is 0
- Find another way of cutting up a 2-holed torus into vertices,
edges
and faces. Show your way and use it to show that the Euler
Characteristic of a 2-holed torus is -2.
Turn in your assignment by
taking a picture of your work. Get your picture onto your
computer, and copy and paste your picture(s) into a Word
document. Save the Word document and put it in the Canvas
Assgnment Euler Characteristic of Tori
Do these on or around April 9
Optional: Video discussion
online at 2:00 April 9th. It will be recorded and available in
Canvas afterwards.
Join Zoom Meeting
https://zoom.us/j/753307733
Meeting ID: 753 307 733
Puzzles and deductions you can make using Euler Characteristic and
other properties of polyhedra
🎥
Video 1: Easy puzzles
Written out easy puzzle example
⋮
Assignment: Do Euler
Characteristic Puzzle Quiz in Canvas
🎥 Video 2: Numberphile--how
many panels on a soccer ball
🎥 Video 3: Harder Euler characteristic deductions/puzzles
The equation sheet from video 3
📜
Assignment:
Pick one of the more regular/uniform polyhedra (one of the Archemedean
Solids) from the Polyhedra
Viewer
Show/explain how to deduce the number of each type of polygon in the
polyhedron in a similar way to those in Videos 2 and 3.
Turn in your assignment by explaining your work in a Word document, and upload it to the Canvas dropbox Polyhedron deductions
🗐
Unit project:
Choose an infinite family of polyhedra, and do all of the following for
that family:
- Use Geogebra to make nets for a member of the family
- Find an equation that will tell you (with the proper input) the
number of faces of each member of the family, and explain how you found
it
- Find an equation that will tell you the number of edges of each
member of the family, and explain how you found it
- Find an equation that will tell you the number of vertices of
each member of the family, and explain how you found it
Suggested families of polyhedra from the Polyhedra
Viewer
- The anti-prisms
- The bi-pyramids
- The elongated bi-pyramids
- The cupolas
- The elongated cupolas
- The gyroelongated cupolas
- The gyrobicupolas
Example: If you were doing
the family of pyramids (not in the list because that's too easy), and N
was the number of sides of the base, then the number of vertices would
be V = N+1, because there are n vertices around the base, and 1 vertex
at the top.
Turn in your project by showing your work in a Word document, and upload it to the Canvas dropbox Polyhedra Project