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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 Eight Star Six Petal Rosette Single Distlefink Mighty Oak Double Eagle Triple Star Double Distlefink

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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 1800’s Region is settled by German immigrants 1850’s Six Petal Rosettes appear 1900’s 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 Has a d6 Finite Rotational Property A Finite Design has 2 classification: “Cyclic” (cn) or “Dihedral” (dn)

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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 One Fold Reflection m1 pm11 pxyz p: pattern x: vertical reflection y: horizontal reflection z: half-turns

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Triple Star NOTE: 20 points on Circle 4 pentagonal star-polygons Order: {5,2} Any {n,k} Star- polygon is made in one stroke

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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 2 nd point

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Construction of the Golden Cut Golden Ratio : AB = AG AG 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

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The Golden Cut on a Circle 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

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10-20 Congruent Points Bisect Angle 1B2 to get a midpoint on arc of Circle B Continue this process around the entire Circle

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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

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CHAPTER 1: Tools of Geometry

CHAPTER 1: Tools of Geometry

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