Geometric Delights : Geometric Delights July 24, 2011
Speaker: Luyi Zhang
Welcome! : Welcome! I’m an MIT student and hopeful math major.
I am enamored by geometry.
I turn simple everyday materials — printer paper, pipe cleaners, yarn, blocks of cheese — into unique and memorable creations.
Slide 3 :
Slide 4 : Geometry
is
delightful.
Overview : Overview A simple trapezoid problem: examining the gap in problem-solving approaches
The importance of creativity in geometry: two challenging examples with simple roots
Exploring the icosahedron: a problem, and a delightful hands-on activity
A simple area problem: trapezoid : A simple area problem: trapezoid
Slide 7 :
Most pre-collegiate math… : Most pre-collegiate math… Heavily reliant on formulas
Students memorize the formulas, often without knowing what they mean or where they came from
This leads to a shallow understanding of math from a young age
A more visual approach: scissors and tape : A more visual approach: scissors and tape We cut the trapezoid into two simpler shapes that we can work with more easily:
SEEing the problem : SEEing the problem
Yet another way of SEEing the problem : Yet another way of SEEing the problem You can rotate a second trapezoid, connect it with the first to create a rectangle!
Problem-Solving Strategies: How was method 1 different from method 2? : Problem-Solving Strategies: How was method 1 different from method 2? Instead of formulas, visualize the problem, SEE what is going on
Break up an unfamiliar problem into familiar parts
Come to understand relationships between components of problem
This understanding is what is most important!
Hard Problems : Hard Problems True problem-solving is not about applying formulas or performing calculations.
Rather, it is about taking simple, fundamental concepts and applying them in creative ways.
One example of a hard
problem: PA = AB = 4;
Find the radius of the small
circle.
Solution outline : Solution outline Similar triangles; orange
and blue
Use pythagorean theorem on
on the right triangle
created by the pink line.
Slide 15 : In a circle, three parallel chords have the following lengths. The chords are a distance of 3 units away from each other. Find a.
Slide 16 : Draw a diameter (green line) and use Power of a Point
Solutions: Simple strategies at heart : Solutions: Simple strategies at heart Similar triangles; Pythagorean theorem
Power of a point: ab = cd
A Challenge: USA Math Talent Search Problem : A Challenge: USA Math Talent Search Problem 5/2/19. Faces ABC and XYZ of a regular icosahedron are parallel, with the vertices labeled such that AX, BY, and CZ are concurrent. Let S be the solid with faces ABC, AYZ, BXZ, CXY, XBC, YAC, ZAB, and XYZ. If AB = 6, what is the volume of S?
Exploring the icosahedron : Exploring the icosahedron 20 triangular faces; 30 edges.
How can we create this icosahedron?
Slide 20 :
Understanding the problem : Understanding the problem The problem asks us to find the volume of ABCDEF, the prism created by two triangles on opposite faces.
Relativity : Relativity Strategy: relate our unknown volume to KNOWN volumes. Namely, that of a hexagonal prism!
We can subtract the volumes of several tetrahedra from the green hexagonal prism.
Slicing off tetrahedra – a cheesy visual : Slicing off tetrahedra – a cheesy visual We slice off 6 congruent tetrahedra in this manner, to yield our desired polyhedron ABCDEF.
Doing the math : Doing the math
Slide 25 : For any “funky polyhedron” truncated from a hexagonal prism like the one above, we have
Volume = a2h √3,
where a is the side length of the prism base and h is the height.
For this problem, a=2√3 and h=(√3)(3+√5), so my answer came out to be 36(3+√5) = 108+36√5.
For a detailed description with calculations, go to http://geometricdelights.wordpress.com/2011/04/17/usamts-cheese
Slide 26 : My website:
http://geometricdelights.wordpress.com
What should we take away from this? : What should we take away from this? At its heart, geometry is built from simple, elegant ideas that are so beautifully intuitive.
Geometry encourages play! Explore, cut-and-paste, make models, and you will reach a deeper, truer understanding.
The ultimate goal is to bring enjoyment of geometric patterns to everyone. Geometry can be so joyful – and with my work, I hope to inspire that joy in others.
Slide 28 :
Slide 29 : Thank you.