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A Geometric Inspection of Pennsylvanian Dutch Hex Signs Evans Evans.

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Presentation on theme: "A Geometric Inspection of Pennsylvanian Dutch Hex Signs Evans Evans."— Presentation transcript:

1 A Geometric Inspection of Pennsylvanian Dutch Hex Signs Evans Evans

2 Common Hex Signs

3 Eight Pointed Star

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

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

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

7 Map of “Dutch Country”

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

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

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

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

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

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

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

15 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

16 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

17 10-20 Congruent Points Bisect Angle 1B2 to get a midpoint on arc of Circle B Continue this process around the entire Circle

18 Pentagonal Star Polygon

19 Quad Star Polygon

20 Shading of the Star

21 Simple Tools

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

23 Double Trinity Tulip One Fold Reflection m1 Inner Circle has a d6 rotational property

24 Where can YOU find a connection between MATH and ART? Any Questions?

25 Thank You A Geometric Inspection of Pennsylvanian Dutch Hex Signs Evan Evans

26 Rosettes in Egypt Rosettes in Asia Rosette in Ancient Rome

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

28 Kissing Circles


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