A Geometric Inspection of Pennsylvanian Dutch Hex Signs

A Geometric Inspection of A Geometric Inspection of Pennsylvanian Dutch Hex SignsPennsylvanian Dutch Hex Signs

Evans Evans Evans Evans

Common Hex SignsCommon Hex Signs

Eight Pointed StarEight Pointed Star

Common Hex SignsCommon Hex Signs

Eight StarSix Petal Rosette

Single DistlefinkMighty Oak

Double Eagle

Triple StarDouble Distlefink

What is a Hex SignWhat is a Hex Sign

Bright Colorful Geometric DesignsBright Colorful Geometric Designs

Circular discs painted on barnsCircular discs painted on barns

Star-Polygons, Birds, Tulips, & HeartsStar-Polygons, Birds, Tulips, & Hearts

Where are they?Where are they?

““Pennsylvania Dutch Country”Pennsylvania Dutch Country”

Southeastern PennsylvaniaSoutheastern Pennsylvania

Between Philadelphia and HarrisburgBetween Philadelphia and Harrisburg

Map of “Dutch Country”Map of “Dutch Country”

When it all BeganWhen it all Began

1800’s Region is settled by German 1800’s Region is settled by German immigrants immigrants

1850’s Six Petal Rosettes appear1850’s Six Petal Rosettes appear

1900’s Birds, Hearts, Tulips appear1900’s Birds, Hearts, Tulips appear

How they Do That?How they Do That?

Simple ToolsSimple Tools

Nail, Rope, Straight Edge, PaintNail, Rope, Straight Edge, Paint

Creativity and Ingenuity Creativity and Ingenuity

Six Petal RosetteSix Petal Rosette

Has a d6 Finite Rotational Property

A Finite Design has 2 classification:

“Cyclic” (cn) or

“Dihedral” (dn)

Construction of a RosetteConstruction of a Rosette

A BA BA BA BA BA BA B

Congruent Circle with center at Point BAnother Congruent Circle with center at new intersection point

Double DistlefinkDouble Distlefink

One Fold Reflection

m1

pm11

pxyz

p: pattern

x: vertical reflection

y: horizontal reflection

z: half-turns

Triple StarTriple Star

NOTE:20 points on Circle 4 pentagonal

star-polygons

Order: {5,2}

Any {n,k} Star-polygon is made in one stroke

Star-PolygonStar-Polygon

Pentagonal Star-Polygon

{5,2}

5

1

2

3

45

1

2

3

45

1

2

3

45

1

2

3

45

1

2

3

45

1

2

3

4

Divide Circle into 5 equally spaced points

Start at point “1” and connect every 2nd point

Construction of the Golden CutConstruction of the Golden Cut

CA BCA B

D

CA B

D

CA B

ED

CA BG

ED

CA B

Golden Ratio : AB = AGAG GB

Phi :

Line Segment AB with Midpoint C

Construct a Perpendicular Line to AB through point B

Make Circle B with radius BC to find Point D on perpendicular line

Form Triangle ABD

Make Circle D

with radius DB to find

point E on line AD

Make Circle A

with radius AE to find

point G on line segment AB

The Golden Cut on a CircleThe Golden Cut on a Circle

GA BGA BGA BGA BGA BGA BGA B

Make Circle B

with radius AB

Construct Circle A

with radius AG

Construct another Circle with radius AG and center at intersection of the 2 Circles

Repeat this process around the circumference of the Circle B

Finish with exactly 10 Circles

and 10 equally spaced points

around the Center Circle B

10-20 Congruent Points10-20 Congruent Points

9 8

7

6

5

43

10

2

G1

B

9 8

7

6

5

43

10

2

G

1B

9 8

7

6

5

43

10

2

G

1B

1816

14

12

10

86

20

4

2

17 15

13

11

9

75

19

3

G

1B

Bisect Angle 1B2 to get a midpoint on arc of Circle B

Continue this process around the entire Circle

Pentagonal Star PolygonPentagonal Star Polygon

1

2

3

5

4

1

2

3

5

4

1

2

3

5

45

1

1

2 2

3

3

5

4

4

5

1

1

2 2

3

3

5

4

45

1

2

3

4

5

1

1

2 2

3

3

5

4

45

1

2

3

4

5

1

1

2 2

3

3

5

4

4

5

5

1

12

2

3

4

3

4

5

1

1

2 2

3

3

5

4

4

5

5

1

12

2

3

4

3

4

5

1

1

2 2

3

3

5

4

4

Quad Star PolygonQuad Star Polygon

Shading of the StarShading of the Star

BAA BBAA BBA

Simple Tools

Maple LeafMaple LeafOuter Ring has a c1 finite design

Inner Ring has a d8 rotational property

Double Trinity TulipDouble Trinity Tulip

One Fold Reflection m1

Inner Circle has a d6 rotational property

Where can YOU

find a connection between

MATH and ART?

Any Questions?

Thank YouThank You

A Geometric Inspection of A Geometric Inspection of Pennsylvanian Dutch Hex SignsPennsylvanian Dutch Hex Signs

Evan EvansEvan Evans

Rosettes in EgyptRosettes in AsiaRosette in Ancient Rome

Daddy HexDaddy Hex

Outer Ring is a c4’ finite rotation

Inner Ring is a d2’ finite rotation

A BA BA BA B

Kissing Circles