Miles of Tiles The Exhibit Miles of Tiles The Exhibit.

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Presentation transcript:

Miles of Tiles The Exhibit

Miles of Tiles The Exhibit

Miles of Tiles The Exhibit

Miles of Tiles The Exhibit

Miles of Tiles The Exhibit

The concept of repeated shapes touches all disciplines.

ART

M. C. Escher

ART

SCIENCE Geology

SCIENCE Geology Sodium Chloride

SCIENCE Geology Sodium Chloride

SCIENCE Geology Sodium Chloride

SCIENCE Geology Galena

SCIENCE Geology Galena

SCIENCE Geology Calcite

SCIENCE Geology Calcite

SCIENCE Biology

SCIENCE Biology Influenza

SCIENCE Biology Influenza

SCIENCE Biology HIV

SCIENCE Biology HIV

SCIENCE Biology T4 bacteriophage

SCIENCE Biology Ingest Nutrients Respiration Excretion

SCIENCE Biology Alive

SCIENCE Biology It is the virus’ shape (structure) that lets it replicate

SCIENCE Biology

SCIENCE Biology Only 4 repeated nucleotides make an entire human being

SCIENCE Architecture

SCIENCE Architecture

SCIENCE Architecture

SCIENCE Architecture

SCIENCE Nature

SCIENCE Nature

SCIENCE Nature

Math Geometry

Math Geometry

Math Geometry

Math Geometry

Math Geometry

Math Geometry + Nature

Math Geometry + Nature

Math Geometry + Nature

Math Geometry + Nature Why do bees use hexagons? Is there something special about them?

Math Geometry + Nature Why do bees use hexagons? Why not build using some other shape?

Math Geometry + Nature Why not build using some other shape? +

Math Geometry + Nature Why not build using some other shape? + Some shapes don’t tessellate!

Math Geometry + Nature Some shapes don’t tessellate! Tessellate? What’s that?

Math Geometry + Nature Tessellated [tes-uh-ley-ted] adjective 1.of, pertaining to, or like a mosaic. 2.arranged in or having the appearance of a mosaic

Math Geometry + Nature Some shapes don’t tessellate! + The bees would have wasted space in the hive

Math Geometry + Nature + The bees would have wasted space in the hive

Math Geometry + Nature But many shapes do tessellate!

Math Geometry + Nature But many shapes do tessellate! Squares seem to be an obvious choice

Math Geometry + Nature They are simple and don’t waste hive space Squares seem to be an obvious choice

Math Geometry + Nature So why do bees choose the more complex shape of a hexagon?

Math Geometry + Nature So why do bees choose the more complex shape of a hexagon? For the answer we’ll need to use some math

Math Geometry + Nature For the answer we’ll need to use some math

Math Geometry + Nature Let’s see what happens when the bees create their hive using different shapes.

Math Geometry + Nature We’ll start with the seemingly simplest design; a square.

Math Geometry + Nature Let’s assume each side of the square is three units in length

Math Geometry + Nature This gives us a square with a perimeter of = 12 P = 12

Math Geometry + Nature The area of this square is x 3 = 9 P = 12 A = 9

Math Geometry + Nature Now let’s do the same calculations for a hexagon also with perimeter of 12.

Math Geometry + Nature 2 2 P = Now let’s do the same calculations for a hexagon also with perimeter of 12.

Math Geometry + Nature 2 2 P = 12 A = ? Now let’s calculate the area for this hexagon.

Math Geometry + Nature Divide the hexagon into triangles 2 P = 12 A = ?

Math Geometry + Nature Each is an equilateral triangle P = 12 A = ? 2

Math Geometry + Nature Calculate area of one equilateral triangle 22 P = 12 A = ? 2

Math Geometry + Nature Area = ½ b x h 22 P = 12 A = ? 2

Math Geometry + Nature A = ½ b x h 22 P = 12 A = ? 2

Math Geometry + Nature A = ½ b x h 22 P = 12 A = ? 11

Math Geometry + Nature A = ½ b x h 2 P = 12 A = ? 1 h

Math Geometry + Nature A = ½ b x h 2 P = 12 A = ? 1 h Determine the value of “h”

Math Geometry + Nature Word of the Day apothem - a perpendicular from the center of a regular polygon to one of its sides.

Math Geometry + Nature A = ½ b x h 2 P = 12 A = ? 1 h Determine the value of “h”

Math Geometry + Nature a 2 + b 2 = c 2 2 P = 12 A = ? 1 h Determine the value of “h”

Math Geometry + Nature h 2 = P = 12 A = ? 1 h Determine the value of “h”

Math Geometry + Nature h 2 = 4 2 P = 12 A = ? 1 h Determine the value of “h”

Math Geometry + Nature h 2 = 3 2 P = 12 A = ? 1 h Determine the value of “h”

Math Geometry + Nature h 2 = 3 2 P = 12 A = ? 1 h Determine the value of “h”

Math Geometry + Nature h = 3 2 P = 12 A = ? 1 h Determine the value of “h”

Math Geometry + Nature h ≈ P = 12 A = ? 1 h ≈ 1.73 Determine the value of “h”

Math Geometry + Nature A = ½ b x h P = 12 A = ? 11 h Now we can calculate the area of one equilateral triangle

Math Geometry + Nature A = 1 x 1.73 P = 12 A = ? 11 h Now we can calculate the area of one equilateral triangle

Math Geometry + Nature A = 1.73 P = 12 A = ? The area of one equilateral triangle is

Math Geometry + Nature Now calculate the area of the hexagon P = 12 A = ? 1.73

Math Geometry + Nature Now calculate the area of the hexagon P = 12 A = ? 1.73

Math Geometry + Nature A=1.73 x 6 P = 12 A = ? 1.73

Math Geometry + Nature A = P = 12 A =

Math Geometry + Nature Let’s do the same calculations one last time for a circle also with perimeter of 12. P = 12 A = ?

Math Geometry + Nature C = ∏ x d P = 12 A = ?

Math Geometry + Nature 12 = ∏ x d P = 12 A = ?

Math Geometry + Nature 12/∏ = d P = 12 A = ?

Math Geometry + Nature 3.82 ≈ d P = 12 A = ?

Math Geometry + Nature 3.82 ≈ d r = ½ d P = 12 A = ?

Math Geometry + Nature 3.82 ≈ d r = ½ 3.82 P = 12 A = ?

Math Geometry + Nature 3.82 ≈ d r = 1.91 P = 12 A = ?

Math Geometry + Nature r = 1.91 P = 12 A = ?

Math Geometry + Nature r = 1.91 A = ∏ r 2 P = 12 A = ?

Math Geometry + Nature r = 1.91 A = ∏ P = 12 A = ?

Math Geometry + Nature r = 1.91 A = ∏ 3.65 P = 12 A = ?

Math Geometry + Nature r = 1.91 A = P = 12 A = 11.47

Math Geometry + Nature Let’s compare what we calculated P = 12 A = 9 P = 12 A = P = 12 A = 11.47

Math Geometry + Nature P = 12 A = 9 If bees built their hive using squares they would have used 12 units of wax to create the walls of each cell, and gotten 9 units of area inside.

Math Geometry + Nature If they build using hexagons they would use 12 units of wax to create the walls of each cell, to get units of area inside. P = 12 A = 10.38

Math Geometry + Nature In other words, they get a larger interior for each cell for the same amount of wax when using hexagons. P = 12 A = 10.38

Math Geometry + Nature If they build using circles they would use 12 units of wax to create the walls of each cell, to get units of area inside. P = 12 A = 11.47

Math Geometry + Nature A circle yields the greatest interior space for the smallest perimeter of any shape! P = 12 A = 11.47

Math Geometry + Nature But circles don’t tessellate! P = 12 A =

Math Geometry + Nature P = 12 A = Hexagons yields the greatest interior space for the smallest perimeter of any shape that tessellates!

Math Geometry + Nature P = 12 A = How do the bees know this?

Miles of Tiles Going Further Penrose tessellations

Miles of Tiles Going Further Penrose tessellations