A Geometric Inspection of Pennsylvanian Dutch Hex Signs

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A Geometric Inspection of Pennsylvanian Dutch Hex
Signs

• Evans Evans

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Common Hex Signs
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Eight Pointed Star
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Common Hex Signs
Double Eagle
Six Petal Rosette
Eight Star
Double Distlefink
Triple Star
Single Distlefink
Mighty Oak
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What is a Hex Sign

• Bright Colorful Geometric Designs
• Circular discs painted on barns
• Star-Polygons, Birds, Tulips, Hearts

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Where are they?

• Pennsylvania Dutch Country
• Southeastern Pennsylvania
• Between Philadelphia and Harrisburg

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Map of Dutch Country
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When it all Began

• 1800s Region is settled by German immigrants
• 1850s Six Petal Rosettes appear
• 1900s Birds, Hearts, Tulips appear

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How they Do That?

• Simple Tools
• Nail, Rope, Straight Edge, Paint
• Creativity and Ingenuity

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Six Petal Rosette
A Finite Design has 2 classification Cyclic
(cn) or Dihedral (dn)
Has a d6 Finite Rotational Property
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Construction of a Rosette
Congruent Circle with center at Point B
Another Congruent Circle with center at new
intersection point
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Double Distlefink
pxyz p pattern x vertical reflection y
horizontal reflection z half-turns
One Fold Reflection
pm11
m1
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Triple Star
Any n,k Star-polygon is made in one stroke
NOTE20 points on Circle
4 pentagonal star-polygons Order 5,2
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Star-Polygon
Pentagonal Star-Polygon 5,2
Divide Circle into 5 equally spaced points
Start at point 1 and connect every 2nd point
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Construction of the Golden Cut
Construct a Perpendicular Line to AB through
point B
Line Segment AB with Midpoint C
Make Circle B with radius BC to find Point D on
perpendicular line
Golden Ratio
Form Triangle ABD
Make Circle A with radius AE to find point G on
line segment AB
AB AG AG GB
Make Circle D with radius DB to find point E on
line AD
Phi
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The Golden Cut on a Circle
Construct Circle A with radius AG
Finish with exactly 10 Circles and 10 equally
spaced points around the Center Circle B
Make Circle B with radius AB
Construct another Circle with radius AG and
center at intersection of the 2 Circles
Repeat this process around the circumference of
the Circle B
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10-20 Congruent Points
Continue this process around the entire Circle
Bisect Angle 1B2 to get a midpoint on arc of
Circle B
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Pentagonal Star Polygon
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Quad Star Polygon
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Shading of the Star
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Simple Tools
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Maple Leaf
Outer Ring has a c1 finite design
Inner Ring has a d8 rotational property
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Double Trinity Tulip
One Fold Reflection m1
Inner Circle has a d6 rotational property
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Where can YOU find a connection between MATH
and ART?
Any Questions?
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Thank You

• A Geometric Inspection of Pennsylvanian Dutch Hex
  Signs
• Evan Evans

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Rosettes in Egypt
Rosettes in Asia
Rosette in Ancient Rome
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Daddy Hex
Outer Ring is a c4 finite rotation
Inner Ring is a d2 finite rotation
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Kissing Circles