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

2/8

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

8/14/2019 Cool investigations in geometry

3/8

2011 Michael Serra 3

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?

8/14/2019 Cool investigations in geometry

4/8

2011 Michael Serra 4

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.

8/14/2019 Cool investigations in geometry

5/8

2011 Michael Serra 5

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

8/14/2019 Cool investigations in geometry

6/8

2011 Michael Serra 6

8/14/2019 Cool investigations in geometry

7/8

2011 Michael Serra 7

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 .

8/14/2019 Cool investigations in geometry

8/8

2011 Michael Serra 8

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?