# Dodecagon

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A Dodecagram is a polygon with twelve edges and angles

• Polygrams
 12-1 12-2 12-3 12-4 12

• Petrie Polygons

A11

 11-simplex Rectified 11-simplex Birectified 11-simplex Trirectified 11-simplex Quadrirectified 11-simplex Quintirectified 11-simplex

BC6

 6-orthoplex Rectified 6-orthoplex Birectified 6-orthoplex Birectified 6-cube Rectified 6-cube 6-cube

D7

 t5(141) t4(141) t3(141) t2(141) t1(141) t0(141)

E6

 t0(221) t1(221) t1(122) t0(122)

F4

 24-cell Rectified 24-cell Snub 24-cell

• Information
• The Dodecagon has 12 Edges and Verticies
• Its Schläfli Symbol is {12}
• Its Coxeter Diagram is
• Its Symmetry Group is D12, order 2x12
• cyclic, convex, equilateral, isogonal, isotoxal
• $A=6\sin\left(\frac{\pi}{6}\right)r^2=3r^2$ (Where $r$ is the radius of the circumscribed circle.)
• One internal is 150°
• All internal angles added up is 1,800°
• Dual Polygon: Itself
• Propertires: convex, cyclic, equilateral, isogonal, isotoxal
• 54 Diagonals and 10 different triangles made from diagonals

• Real Life Examples
• In block capitals, the letters H, X and E(and I in a slab serif font) have dodecagonal outlines.
• The regular dodecagon features prominently in many buildings. The Torre del Oro is a dodecagonal military watchtower in Seville, southern Spain, built by the Almohad dynasty. The early thirteenth century Vera Cruz church in Segovia, Spain is dodecagonal. Another example is the Porta di Venere (Venus' Gate), in Spello, Italy, built in the 1st century BC has two dodecagonal towers, called "Propertius' Towers".
• Regular dodecagonal coins include:
• British threepenny bit from 1937 to 1971, when it ceased to be legal tender.
• British One Pound Coin to be introduced in 2017.
• Australian 50-cent coin
• Fijian 50 cents
• Tongan 50-seniti, since 1974
• Solomon Islands 50 cents
• Croatian 25 kuna
• Romanian 5000 lei, 2001–2005
• South Vietnamese 25 đồng, 1968–1975
• Zambian 50 ngwee, 1969–1992
• Malawian 50 tambala, 1986–1995
• Mexican 20 centavos, since 1992

• Teselation
• Four Examples

• Construction
• Use a compass to draw a circle
• Draw a pair of perpendicular lines that meet at the center of the circle. One horizontal and one verticle.
• Make two circle that center at the two spots where the verticle line intersects the circle. The curve of the two circles have to meet at the center of the original circle.
• Make two circle that center at the two spots where the horizontal line intersects the circle. The curve of the two circles have to meet at the center of the circle you made in step one.
• Starting at the top of the circle, where the verticle line meets the curve, draw a line from there to the nearest point where the large circle's curve intersects with a line or another curve.
• Draw another line from there to the the next intersection along the curve and keep on doing that in a clockwise(or counter-clockwise) direction.
• Erase all lines and curves except the ones made in the past 2 steps.

• Graphing
• If a regular dodecagram with each side having 2 units long was graphed with the center at (0,0), the coordinate would be:
• $(2+\sqrt3,-1)$
• $(1+\sqrt3,-1-\sqrt3)$
• $(1,-2-\sqrt3)$
• $(-1,-2-\sqrt3)$
• $(-1-\sqrt3,-1-\sqrt3)$
• $(-2-\sqrt3,-1)$
• $(-2-\sqrt3,1)$
• $(-1-\sqrt3,1+\sqrt3)$
• $(-1,2+\sqrt3)$
• $(1,2+\sqrt3)$
• $(1+\sqrt3,1+\sqrt{3})$
• $(2+\sqrt3,1)$

• Non-Euclidean Geometry
• Dodecagonal Antiprism