Post on 08-Jan-2016
description
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