|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| {{Even polygon db|Even polygon stat table|p12}}
| |
| In [[geometry]], a '''dodecagon''' is any [[polygon]] with [[12 (number)|twelve]] sides and twelve [[angle]]s.
| |
|
| |
|
| ==Regular dodecagon==
| |
| A [[regular polygon|regular]] dodecagon has all sides of equal length and all angles equal to 150°. It has 12 lines of symmetry and rotational symmetry of order 12. Its [[Schläfli symbol]] is {12}.
| |
|
| |
|
| The [[area]] of a regular dodecagon with side ''a'' is given by:
| | How important are jewelry items for a woman? When men go hunting for presents possess no idea that we for you to most beautiful and superb pieces of accessories. Now you can also be one men guys who pay awareness to a woman's gift. Which are have turn out to be expensive, we must put your basis and chose the most appealing Cheap Silver plate Pendants.<br><br><br><br>Diamond jewellery is a usually thought of gift using best purposes behind the idea. It is clearly evident that, sale of jewelry has been growing like any thing since so few years. Diamond jewelry UK mainly focused to cater you with lot of varieties in diamond earrings. Good amount is certainly best substitute for invest on this purchase. Should you want to save profit prior then, Diamond jewelry UK works to you. You will see many people go for this pick.<br><br>February - Purple Amethyst: This stone represents a personality which exemplifies delicate. You are at peace and keep the same to others. The said pertaining to being a very faithful person with a pure and modest connected with living each day.<br><br>Many epidermis trendy silver Pendants are available various different designs. Women are choosing the glamour look and also pendant made of silver usually goes anything at all. Modern designs are simple a single will get different epidermis geometric information. The figures are arranged differently and yes it gives the pendant fresh look. On the other hand, 1 wants attempt to for a hard-to-find look, then opting for such Pendants is a good understanding. One can try special designs in pieces of Celtic and Gothic design. They are very much in fashion and many people are choosing the group. At the same time, one additionally try for pendants a variety of shapes of animals and also letters.<br><br>There is one problem in this particular scenario. You may not want to make use of your glasses every time you dine out or have that compares any list like the films. It can be awkward socially.<br><br>Hats often in trend in the wintertime season. Huge interwoven beanies and oversized Moroccan furry hats are good to have this freezing. It has never been from the style typically the winter process.<br><br>Better to prefer cost card over debit card. It keeps you away through the cheating. If you have had one-time virtual plastic money, it get best a person personally. Moreover, go through the privacy, refund, exchange, shipping statement. Thoroughly read the site so that you do not have pay out for the hefty and hidden amount for the additional benefits that you don't want to reach all.<br><br>Whatever gemstone you choose it are going to a stellar addition to enhance an outfit and daily life you are leading. Gemstone rings, bracelets, pendants and bracelets are a definite great gift for the special women in your own. Consider a behavior which stands in order to you as representing the buyer that get the gift. Gemstone jewelry can be a guaranteed safe bet.<br><br>If you have any inquiries pertaining to where and how to use [http://m.scalarenergypendants.com/ Quantum Science Pendant], you can make contact with us at the web-site. |
| :<math>\begin{align} A & = 3 \cot\left(\frac{\pi}{12} \right) a^2 =
| |
| 3 \left(2+\sqrt{3} \right) a^2 \\
| |
| & \simeq 11.19615242\,a^2.
| |
| \end{align}</math>
| |
| | |
| Or, if ''R'' is the radius of the [[circumscribe]]d circle,<ref>See also [[József Kürschák|Kürschák]]'s geometric proof on [http://demonstrations.wolfram.com/KurschaksDodecagon/ the Wolfram Demonstration Project]</ref>
| |
| :<math>A = 6 \sin\left(\frac{\pi}{6}\right) R^2 = 3 R^2.</math>
| |
| | |
| And, if ''r'' is the radius of the [[inscribe]]d circle,
| |
| :<math>\begin{align} A & = 12 \tan\left(\frac{\pi}{12}\right) r^2 =
| |
| 12 \left(2-\sqrt{3} \right) r^2 \\
| |
| & \simeq 3.2153903\,r^2.
| |
| \end{align}</math>
| |
| | |
| A simple formula for area (given the two measurements) is: <math>\scriptstyle A\,=\,3ad</math> where ''d'' is the distance between parallel sides.
| |
| | |
| Length ''d'' is the height of the dodecahedron when it sits on a side as base, and the diameter of the inscribed circle.
| |
| | |
| By simple trigonometry, <math>\scriptstyle d\,=\,a(1\,+\,2cos{30^\circ}\,+\,2cos{60^\circ})</math>.
| |
| | |
| == Uses ==
| |
| A regular dodecagon can fill a plane vertex with other regular polygons:
| |
| {| class=wikitable
| |
| |[[File:3.12.12 vertex.png|120px]]<BR>3.12.12
| |
| |[[File:4.6.12 vertex.png|120px]]<BR>4.6.12
| |
| |[[File:3.3.4.12 vertex.png|120px]]<BR>3.3.4.12
| |
| |[[File:3.4.3.12 vertex.png|120px]]<BR>3.4.3.12
| |
| |}
| |
| | |
| ==Dodecagon construction==
| |
| A regular dodecagon is [[constructible polygon|constructible]] using [[compass and straightedge]]:
| |
| | |
| [[File:Regular Dodecagon Inscribed in a Circle.gif]]<br>Construction of a regular dodecagon
| |
| | |
| ==Occurrence==
| |
| ===Tiling===
| |
| Here are 3 example [[Tiling by regular polygons|periodic plane tilings]] that use dodecagons:
| |
| {| width=640 class="wikitable"
| |
| |[[Image:Tile 3bb.svg|205px|Tile 3bb.svg]]<br>[[Truncated hexagonal tiling|Semiregular tiling 3.12.12]]
| |
| |[[Image:Tile 46b.svg|205px]]<br>[[Great rhombitrihexagonal tiling|Semiregular tiling: 4.6.12]]
| |
| |[[Image:Dem3343tbc.gif|205px]]<br>A [[Tilings of regular polygons#Other edge-to-edge tilings|demiregular tiling]]:<br>3.3.4.12 & 3.3.3.3.3.3
| |
| |}
| |
| ===Pattern blocks===
| |
| [[File:Wooden pattern blocks dodecagon.JPG||thumb|Dodecagon made with [[pattern blocks]]]]
| |
| One of the ways the [[mathematical manipulative]] [[pattern blocks]] are used is in creating a number of different dodecagons.<ref>"Doin' Da' Dodeca'" on [http://mathforum.org/berman/patternblocks/classics/responses/chal2/doin_da_dodeca.html mathforum.org]</ref>
| |
| ===Petrie polygons===
| |
| The regular dodecagon is the [[Petrie polygon]] for many higher dimensional polytopes, seen as [[orthogonal projection]]s in [[Coxeter plane]]s, including:
| |
| | |
| {| class=wikitable width=540
| |
| |- align=center valign=top
| |
| !valign=center|A<sub>11</sub>
| |
| |[[File:11-simplex_t0.svg|100px]]<br>[[11-simplex]]
| |
| |[[File:11-simplex_t1.svg|100px]]<br>[[Rectified 11-simplex]]
| |
| |[[File:11-simplex_t2.svg|100px]]<br>[[Birectified 11-simplex]]
| |
| |[[File:11-simplex_t3.svg|100px]]<br>[[Trirectified 11-simplex]]
| |
| |[[File:11-simplex_t4.svg|100px]]<br>[[Quadrirectified 11-simplex]]
| |
| |[[File:11-simplex_t5.svg|100px]]<br>[[Quintirectified 11-simplex]]
| |
| |- align=center valign=top
| |
| !valign=center|BC<sub>6</sub>
| |
| |[[File:6-cube_t5.svg|100px]]<br>[[6-orthoplex]]
| |
| |[[File:6-cube_t4.svg|100px]]<br>[[Rectified 6-orthoplex]]
| |
| |[[File:6-cube_t3.svg|100px]]<br>[[Birectified 6-orthoplex]]
| |
| |[[File:6-cube_t2.svg|100px]]<br>[[Birectified 6-cube]]
| |
| |[[File:6-cube_t1.svg|100px]]<br>[[Rectified 6-cube]]
| |
| |[[File:6-cube_t0.svg|100px]]<br>[[6-cube]]
| |
| |- align=center valign=top
| |
| !valign=center|D<sub>7</sub>
| |
| |[[File:7-cube_t6_B6.svg|100px]]<br>[[7-orthoplex|t<sub>5</sub>(1<sub>41</sub>)]]
| |
| |[[File:7-cube_t5_B6.svg|100px]]<br>[[Rectified 7-orthoplex|t<sub>4</sub>(1<sub>41</sub>)]]
| |
| |[[File:7-cube_t4_B6.svg|100px]]<br>[[Birectified 7-orthoplex|t<sub>3</sub>(1<sub>41</sub>)]]
| |
| |[[File:7-cube_t3_B6.svg|100px]]<br>[[Birectified 7-cube|t<sub>2</sub>(1<sub>41</sub>)]]
| |
| |[[File:7-demicube_t1_D7.svg|100px]]<br>[[Rectified 7-cube|t<sub>1</sub>(1<sub>41</sub>)]]
| |
| |[[File:7-demicube_t0_D7.svg|100px]]<br>[[7-demicube|t<sub>0</sub>(1<sub>41</sub>)]]
| |
| |- align=center valign=top
| |
| !valign=center|[[E6 (mathematics)|E<sub>6</sub>]]
| |
| |[[File:E6 graph.svg|100px]]<br>[[2 21 polytope|t<sub>0</sub>(2<sub>21</sub>)]]
| |
| |[[File:Up 2 21 t1 E6.svg|100px]]<br>t<sub>1</sub>(2<sub>21</sub>)
| |
| |[[File:Up 2 21 t2 E6.svg|100px]]<br>t<sub>1</sub>(1<sub>22</sub>)
| |
| |[[File:Gosset 1 22 polytope.png|100px]]<br>[[1 22 polytope|t<sub>0</sub>(1<sub>22</sub>)]]
| |
| |- align=center valign=top
| |
| !valign=center|[[F4 (mathematics)|F<sub>4</sub>]]
| |
| |[[File:24-cell_t0_F4.svg|100px]]<br>[[24-cell]]
| |
| |[[File:24-cell_t1_F4.svg|100px]]<br>[[Rectified 24-cell]]
| |
| |[[File:24-cell h01 F4.svg|100px]]<br>[[Snub 24-cell]]
| |
| |}
| |
| | |
| ==Examples in use==
| |
| In [[block capitals]], the letters [[E]], [[H]] and [[X]] (and [[I]] in a [[slab serif]] font) have dodecagonal outlines.
| |
| [[File:Segovia Vera Cruz.jpg|thumb|The Vera Cruz church in [[Segovia]]]]
| |
| The regular dodecagon features prominently in many buildings. The [[Torre del Oro]] is a dodecagonal military [[Watchtower (fortification)|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".
| |
| [[File:1942 threepence reverse.jpg|thumb|A 1942 British threepence, reverse]]
| |
| Regular dodecagonal coins include:
| |
| *[[Threepence (British coin)|British threepenny bit]] from 1937 to 1971, when it ceased to be legal tender.
| |
| *[[Australian 50-cent coin]]
| |
| *[[Coins of the Fijian dollar|Fijian 50 cents]]
| |
| *[[Tongan paʻanga|Tongan 50-seniti]], since 1974
| |
| *[[Solomon Islands dollar|Solomon Islands 50 cents]]
| |
| *[[Croatian kuna|Croatian 25 kuna]]
| |
| *[[Romanian leu|Romanian 5000 lei]], 2001–2005
| |
| *[[Penny (Canadian coin)|Canadian penny]], 1982–1996
| |
| *[[South Vietnamese đồng|South Vietnamese 25 đồng]], 1968–1975
| |
| *[[Zambian kwacha|Zambian 50 ngwee]], 1969–1992
| |
| *[[Malawian kwacha|Malawian 50 tambala]], 1986–1995
| |
| *[[Mexican peso|Mexican 20 centavos]], since 1992
| |
| | |
| ==See also==
| |
| *[[Dodecagonal number]]
| |
| *[[Dodecahedron]] – a regular [[polyhedron]] with 12 [[pentagon]]al faces.
| |
| *[[Dodecagram]]
| |
| | |
| ==Notes==
| |
| {{Reflist}}
| |
| | |
| ==External links==
| |
| *{{MathWorld |urlname=Dodecagon |title=Dodecagon}}
| |
| *[http://www.cut-the-knot.org/Curriculum/Geometry/KurschakTile.shtml Kürschak's Tile and Theorem]
| |
| *[http://www.mathopenref.com/dodecagon.html Definition and properties of a dodecagon] With interactive animation
| |
| {{Polygons}}
| |
| | |
| [[Category:Polygons]]
| |
How important are jewelry items for a woman? When men go hunting for presents possess no idea that we for you to most beautiful and superb pieces of accessories. Now you can also be one men guys who pay awareness to a woman's gift. Which are have turn out to be expensive, we must put your basis and chose the most appealing Cheap Silver plate Pendants.
Diamond jewellery is a usually thought of gift using best purposes behind the idea. It is clearly evident that, sale of jewelry has been growing like any thing since so few years. Diamond jewelry UK mainly focused to cater you with lot of varieties in diamond earrings. Good amount is certainly best substitute for invest on this purchase. Should you want to save profit prior then, Diamond jewelry UK works to you. You will see many people go for this pick.
February - Purple Amethyst: This stone represents a personality which exemplifies delicate. You are at peace and keep the same to others. The said pertaining to being a very faithful person with a pure and modest connected with living each day.
Many epidermis trendy silver Pendants are available various different designs. Women are choosing the glamour look and also pendant made of silver usually goes anything at all. Modern designs are simple a single will get different epidermis geometric information. The figures are arranged differently and yes it gives the pendant fresh look. On the other hand, 1 wants attempt to for a hard-to-find look, then opting for such Pendants is a good understanding. One can try special designs in pieces of Celtic and Gothic design. They are very much in fashion and many people are choosing the group. At the same time, one additionally try for pendants a variety of shapes of animals and also letters.
There is one problem in this particular scenario. You may not want to make use of your glasses every time you dine out or have that compares any list like the films. It can be awkward socially.
Hats often in trend in the wintertime season. Huge interwoven beanies and oversized Moroccan furry hats are good to have this freezing. It has never been from the style typically the winter process.
Better to prefer cost card over debit card. It keeps you away through the cheating. If you have had one-time virtual plastic money, it get best a person personally. Moreover, go through the privacy, refund, exchange, shipping statement. Thoroughly read the site so that you do not have pay out for the hefty and hidden amount for the additional benefits that you don't want to reach all.
Whatever gemstone you choose it are going to a stellar addition to enhance an outfit and daily life you are leading. Gemstone rings, bracelets, pendants and bracelets are a definite great gift for the special women in your own. Consider a behavior which stands in order to you as representing the buyer that get the gift. Gemstone jewelry can be a guaranteed safe bet.
If you have any inquiries pertaining to where and how to use Quantum Science Pendant, you can make contact with us at the web-site.