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8/14/2019 Cool investigations in geometry
1/8
2011 Michael Serra
Cool Investigationsin Geometry
CAMT 2011With Michael Serra
8/14/2019 Cool investigations in geometry
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2011 Michael Serra 2
Resources
Pentominoes Discovering Geometry 4th edition, Serra,
pgs: 53, 63, 87, 113 Polyominoes, Solomon Golomb,
Charles Scribners Sons
Archimedean Tilings Discovering Geometry 4th edition, Serra,
pgs: 389-391
Tilings and Patterns , Grunbaum & ShephardW. H. Freeman & Company
Picks Formula Discovering Geometry 4th edition, Serra,
pgs: 446-448
Origamics
Discovering Geometry 5th edition, Serra, Origamics, Mathematical Explorations Through
Paper Folding, by Kazuo Haga
Gothic Geometry Discovering Geometry 4th edition, Serra,
pgs: 273, 280, 329, 353, 510
Sourcebook of Problems for Geometry ,Mabel Sykes, Dale Seymour Publications
Website www.michaelserra.net
A(b,i) = ( )b + ( )i + ( )
(1/2) 5
5/8
5/62/3
1/6 5/24
1/8
1
5/8
3/8
1/21/2 E
D C
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The Pentomino ChallengePentomino Warm-up1. Find all the Pentominoes.2. Which Pentominoes can fold into a box without a lid?
3. Find the letter that represents each Pentomino.
The Pentomino Challenge4. Which of the Pentominoes can tile the plane and what is the fewest number of
colors needed for each tiling?
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Archimedean Tilings
When the same combination of regular polygons meet in the sameorder at each vertex of a tessellation, it is called an Archimedean
tiling or a 1-uniform tiling.The Three Pure tilings
Three 1-uniform tilings
4.8.8 ___.___.___ ___.___.___There are eleven 1-uniform tilings. Find the other five.
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Picks formula
Austrian mathematician Georg Pick discovered a relationship for thearea of figures on a square dot grid. The relationship known as Picks
formula relates the number of interior lattice points and boundarylattice points with the area of the figure.
# of boundary points ( b )
3 4 5 6 7 8
0 1.5
1 2.5
23
4
5 5.5 # o
f i n t e r
i o r p
o i n t s ( i )
Area of the polygon
1
.5
1
A = 1+.5+ 1 = 2.5
4
3 3
1.5
2
A = 12 (1.5+2+3) = 12 6.5 = 5.5
A = 1 + .5 = 1.5
1 .5
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Origamics Puzzle #1
Unfold. Find the measure of each labeled angle.
Origamics Puzzle #2
(1/2) 5
5/8
5/62/3
1/6 5/24
1/8
1
5/
3/8
1/21/2 E
D C
Fold to locatemidpoints E, F
F
C
Bring DAto DC
H
G
F
D
Reflect ! GBH
over GH
H
G
Bring C tomidline EF
G
CF
D
1. Fold so A and B meet, locating point E. 2. Fold so C and E meet.
3. Prove the values shown are correct .
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Gothic GeometryThe builders of the great Gothic cathedrals were talentedgeometers. They solved a number of problems using basicgeometry tools. For example: How do you construct a circletangent to three equilateral Gothic arches? How do you drawcongruent circles each tangent to two others and internallytangent to a given circle?
Gothic Challenge #1:Explain why this construction works.
Gothic Challenge #2:How do you draw n congruent circles each tangent to
two others and internally tangent to a given circle?