to tessellate: to cover a plane with one or more shapes without overlaps or gaps

To tessellate: to cover a plane with one or more shapes without overlaps or To tessellate: to cover a plane with one or more shapes without overlaps or gapsgaps

A Regular Tessellation is when only one regular A Regular Tessellation is when only one regular shape is used. A regular shape has congruent shape is used. A regular shape has congruent sides and angles sides and angles

There are exactly 3 Regular Tessellations:There are exactly 3 Regular Tessellations:

Square, Equilateral Triangle & HexagonSquare, Equilateral Triangle & Hexagon

The grid for each of these tessellations was done in Sketchpad.

Then each student colored her creation by hand, except for Mary who chose to do hers in Paint.

A Semi-regular tessellations is when two or more A Semi-regular tessellations is when two or more regular shapes are used. At each vertex the same regular shapes are used. At each vertex the same arrangement of shapes must exist. arrangement of shapes must exist.

There are exactly 8 semi-regular tessellationsThere are exactly 8 semi-regular tessellations

Each of these grids was also done in Sketchpad. Then the students colored them in Paint.

4.8.8.

CaitlinCaitlin

#1

3.4.6.4.

AyannaAyanna#2

3.4.6.3

Jennifer

CatherineCatherine

3.3.4.3.4.#3

3.3.4.3.4

3.3.4.3.4

VanaVana

3.3.3.4.4.

ShannonShannon

#4

3.6.3.6.

LaurieLaurie

#5

#6

Anne Marie

3.3.3.3.6

#7

3.3.3.3.6

Nora

Tiffany

3.12.12

#8

NEON

DRAGONS

by Elizabeth

Laurie

Ayanna’s Ocean

TiffanyTiffany

Tiffany

Finished by Vana

All of the tessellations done in class were by translation only. They are periodic tessellations. For further explanation go to

http://www.geocities.com/dottie4math/slides.ppt

To learn more about other types go to

http://mathforum.org/sum95/suzanne/tess.intro.html

To explore aperiodic tessellations scroll down to Penrose on the homepage

http://www.funmath.org

Continue with art show.

is a fractal created by connecting the midpoints of a triangle to form another triangle. The original triangle is now composed of 4 congruent triangles – each of which is similar to the original. The process continues by connecting the midpoints of the remaining triangles – not the center one. On and on it goes!

Students created Sierpenski triangles after the study of congruent triangles and after deriving the proof that states – when the midpoints of two sides of a triangle are connected, that segment is parallel to the third side and equal in measure to ½ its length.

While they followed the algorithm, we discussed, after each iteration, how the new triangles related in area and perimeter to the original. We created formulas that allowed us to determine each measure after a given number of steps.

For Therapy, each student had the freedom to decorate her triangle in a creative fashion.

SHANNON

Caitlin

Tiffany

Katie

This one was done with Sketchpad.

It’s picture was taken with a digital camera as it appeared on the computer screen. The image was then inserted into the slide show.

Reverse Coloring

Sierpenski on Calculator

Whose idea was this???

HA-HA-HA-HA-HA

HA-HA-HA-HA! HA-HA-HA!

Fractals in the Lab

Who is Fibonacci?

Fibonacci’s sequence of numbers:

1 1 2 3 5 8 13 21 . . .

What is a Palindrome?

Ex. radar

Fibonacci Palindromic Sequence:

8 5 3 2 1 1 1 2 3 5 8

Making a Palindromic Design:

Divide the sides of a polygon into divisions equal in length with the above sequence. Connect corresponding points on adjacent or opposite sides. Then color every other region.

AND

All snowflakes have a hexagonal shape, but we made 7 sided flakes instead. Our hope is that Mother Nature will show us what true snow- flakes look like. (And she sure did!!!)

The Christmas tree has a maze inside. As with a deductive proof, it is sometimes easier if you look at the end first and work backwards.

The star on top was created by wrapping a strip of paper into a regular pentagon. Before the wrapping began, a wish was written on the back side of the strip.

Mother Nature’s Response

There are five Platonic Solids. Each is created with exactly one regular polygon.

One of these solids is called the icosahedron. It is formed with 20 equilateral triangles. At each vertex five triangles meet. If each vertex is truncated, the resultant solid is composed of hexagons and pentagons. The name of this solid is the truncated icosahedron which is one of the Archimedian Solids. There are 13 of these solids which are composed of more than one regular shape.

I have more fun with mine!

Sir David Brewster (1781-1868)

History of Kaleidoscopes

Kaleidoscopes are created by using mirrors to reflect a design.

However, with Sketchpad, the following were made by rotating a triangular design. This is a better technique when the creations are activated. Activation can only be seen in Sketchpad, so in this slide show there is only a pretty image to represent each creation.

If you have Sketchpad, see how to see the action at the end of this slide show.

Anne Marie

Ayanna

Caitlin

Catherine

CatieCatie

Elizabeth

Jennifer

Mary

Nora

Shannon

Laurie

Katie

Exploring the Mandala

Vana

Caitlin

Catherine

Ayanna

Catie

Jennifer

Laurie

Katie

Math Art

And for those beautiful kaleidoscopesRemember you must have Sketchpad on your computer!