# A Geometric Inspection of Pennsylvanian Dutch Hex Signs Evans Evans.

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A Geometric Inspection of Pennsylvanian Dutch Hex Signs Evans Evans

Common Hex Signs

Eight Pointed Star

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

What is a Hex Sign Bright Colorful Geometric Designs Circular discs painted on barns Star-Polygons, Birds, Tulips, & Hearts

Where are they? “Pennsylvania Dutch Country” Southeastern Pennsylvania Between Philadelphia and Harrisburg

Map of “Dutch Country”

When it all Began 1800’s Region is settled by German immigrants 1850’s Six Petal Rosettes appear 1900’s Birds, Hearts, Tulips appear

How they Do That? Simple Tools Nail, Rope, Straight Edge, Paint Creativity and Ingenuity

Six Petal Rosette Has a d6 Finite Rotational Property A Finite Design has 2 classification: “Cyclic” (cn) or “Dihedral” (dn)

Construction of a Rosette Congruent Circle with center at Point B Another Congruent Circle with center at new intersection point

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

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

Star-Polygon Pentagonal Star-Polygon {5,2} Divide Circle into 5 equally spaced points Start at point “1” and connect every 2 nd point

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

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

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

Pentagonal Star Polygon

Simple Tools

Maple Leaf Outer Ring has a c1 finite design Inner Ring has a d8 rotational property

Double 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 You A Geometric Inspection of Pennsylvanian Dutch Hex Signs Evan Evans

Rosettes in Egypt Rosettes in Asia Rosette in Ancient Rome

Daddy Hex Outer Ring is a c4’ finite rotation Inner Ring is a d2’ finite rotation

Kissing Circles

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