Antiprism: Difference between revisions

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When you were younger, maybe we were one of those lucky persons whom didnt need fat reduction help. Maybe we were chasing kids, functioning at an outside job, cooking, cleaning plus living an active lifestyle. Its not that guys over 50 plus post-menopausal females are not active. But often they are less active. This can be as a result of bodily limitations, such as arthritis, or considering they have really gotten used to a more sedentary life-style in retirement or even when they are nevertheless functioning.<br><br>No matter what you eat, usually observe the part sizes. There is too much of the advantageous thing! Watch out for cream based soups and sauces; instead try tomato-based because a healthy alternative. Fried foods are a big no-no, have the fish and meats baked or grilled! Instead of a baked dessert, have fresh fruit. As another good direction, utilize cooking spray rather of oils.<br><br>I would greatly endorse a body fat check for anyone who wishes to lose fat (fat weight), whom desires to become on top form, or only desires to sustain an already healthy lifestyle. These are typically low-priced, they provide a starting point for your goals, and they provide wise encouragement to keep we on mark. But never, I reiterate never be preoccupied by the readings. Simply keep in your mind which what we see following taking a measuring is an instant inside time. The accurateness originates from we plus what we realize on top of several weeks plus months. The body fat check is a smart computer, nevertheless like any computer, it will only return to you what you place into it. Work out plus dine nourishing - the check is only a device to do the goals inside life.<br><br>The BMI index is broken down into four principal categories. Each category is represents a amount of morbidity and mortality data or wellness risk. If the [http://safedietplans.com/bmi-calculator bmi calculator] reads lower than 18.5 you're considered underweight, when it reads between 18.5 to 25 we are at an optimal weight; when the bmi calculator reads 25 to 30 a are overweight, if it reads 30 or above than you're considered overweight (over 40 morbidly obese). Patients which score above a 30 are a major concern of the health profession. Studies have shown which being obese could have an impact on a health. These same research show how weight is shortening the average life span.<br><br>Burnello and Beatice D'Hombres of the Euopean Commision , Joint Research Center inside Ispra, Italy utilized information within the European Community House Panel, which is a compative dataset, in the study published in The Journal Economics and Human Biology.<br><br>The simplest means to keep track of the calorie and fat consumption for the day is to keep a log! Hide a small notepad and pen inside the purse particularly for this and take it where we go. Write down what you eat for the day. A superb, free site to calculate how several calories are in your food is a part of iGoogle!<br><br>If BMI is not the proper tool, what exactly is? Well, to actually accurately tell what % of the fat is fat, we have to be weighed underwater. That's a hassle, not the type of thing the average individual will do, not regularly. But, with just a tape measure, you are able to calculate a waist-hip ratio. Next we can utilize http://www.healthcalculators.org/calculators/waist_hip.asp to find when it's inside a healthy range. This really is a more sensible measure.
{| class="wikitable" bgcolor="#ffffff" cellpadding="5" align="right" style="margin-left:10px" width="280"
!bgcolor=#e7dcc3 colspan=2|Set of uniform antiprisms
|-
|align=center colspan=2|[[Image:Hexagonal antiprism.png|280px|Hexagonal antiprism]]<br>
|-
|bgcolor=#e7dcc3|Type||[[uniform polyhedron]]
|-
|bgcolor=#e7dcc3|Faces||2 [[polygon|''n''-gon]]s, 2''n'' [[triangle]]s
|-
|bgcolor=#e7dcc3|Edges||4''n''
|-
|bgcolor=#e7dcc3|Vertices||2''n''
|-
|bgcolor=#e7dcc3|[[Vertex configuration]]||3.3.3.''n''
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||s{2,2''n''}<br>sr{2,''n''}<BR>{ } &#x2a02; {n}
|-
|bgcolor=#e7dcc3|[[Coxeter–Dynkin diagram]]s||{{CDD|node_h|2x|node_h|2x|n|node}}<br>{{CDD|node_h|2x|node_h|n|node_h}}
|-
|bgcolor=#e7dcc3|[[List of spherical symmetry groups|Symmetry group]]||D<sub>''n''d</sub>, [2<sup>+</sup>,2''n''], (2*''n''), order 4''n''
|-
|bgcolor=#e7dcc3|[[Point_groups_in_three_dimensions#Rotation_groups|Rotation group]]||D<sub>''n''</sub>, [2,''n'']<sup>+</sup>, (22''n''), order 2''n''
|-
|bgcolor=#e7dcc3|[[Dual polyhedron]]||[[trapezohedron]]
|-
|bgcolor=#e7dcc3|Properties||convex, semi-regular [[vertex-transitive]]
|-
|bgcolor=#e7dcc3|[[Net (polyhedron)|Net]]||[[Image:Generalized antiprisim net.svg|150px]]
|}
In [[geometry]], an ''n''-sided '''antiprism''' is a [[polyhedron]] composed of two parallel copies of some particular ''n''-sided [[polygon]], connected by an alternating band of [[triangle]]s. Antiprisms are a subclass of the [[prismatoid]]s.
 
Antiprisms are similar to [[prism (geometry)|prism]]s except the bases are twisted relative to each other, and that the side faces are triangles, rather than quadrilaterals.
 
In the case of a regular ''n''-sided base, one usually considers the case where its copy is twisted by an angle 180°/''n''. Extra regularity is obtained by the line connecting the base centers being perpendicular to the base planes, making it a '''right antiprism'''. As faces, it has the two [[n-gon|''n''-gonal]] bases and, connecting those bases, 2''n'' isosceles triangles.
 
==Uniform antiprism==
A '''[[Prismatic uniform polyhedron|uniform]] antiprism''' has, apart from the base faces, 2''n'' equilateral triangles as faces. As a class, the uniform antiprisms form an infinite series of vertex-uniform polyhedra, as do the uniform prisms. For {{nowrap|''n'' {{=}} 2}} we have as degenerate case the regular [[tetrahedron]] as a ''digonal antiprism'', and for {{nowrap|''n'' {{=}} 3}} the non-degenerate regular [[octahedron]] as a ''triangular antiprism''.
 
The [[dual polyhedra]] of the antiprisms are the [[trapezohedra]]. Their existence was first discussed and their name was coined by [[Johannes Kepler]].
 
{{UniformAntiprisms}}
 
==Cartesian coordinates==
[[Cartesian coordinates]] for the vertices of a right antiprism with ''n''-gonal bases and isosceles triangles are
:<math>\left( \cos\frac{k\pi}{n}, \sin\frac{k\pi}{n}, (-1)^k h \right)</math>
 
with ''k'' ranging from 0 to 2''n''−1; if the triangles are equilateral,
:<math>2h^2=\cos\frac{\pi}{n}-\cos\frac{2\pi}{n}.</math>
 
==Volume and surface area==
Let ''a'' be the edge-length of a [[uniform polyhedron|uniform]] antiprism. Then the volume is
:<math>V = \frac{n \sqrt{4\cos^2\frac{\pi}{2n}-1}\sin \frac{3\pi}{2n} }{12\sin^2\frac{\pi}{n}} \; a^3</math>
 
and the surface area is
:<math>A = \frac{n}{2} ( \cot{\frac{\pi}{n}} + \sqrt{3}) a^2.</math>
 
==Symmetry==
The [[symmetry group]] of a right ''n''-sided antiprism with regular base and isosceles side faces is D<sub>''n''d</sub> of order 4''n'', except in the case of a tetrahedron, which has the larger symmetry group T<sub>d</sub> of order 24, which has three versions of D<sub>2d</sub> as subgroups, and the octahedron, which has the larger symmetry group O<sub>h</sub> of order 48, which has four versions of D<sub>3d</sub> as subgroups.
 
The symmetry group contains [[inversion in a point|inversion]] [[if and only if]] ''n'' is odd.
 
The [[rotation group SO(3)|rotation group]] is D<sub>''n''</sub> of order 2''n'', except in the case of a tetrahedron, which has the larger rotation group T of order 12, which has three versions of D<sub>2</sub> as subgroups, and the octahedron, which has the larger rotation group O of order 24, which has four versions of D<sub>3</sub> as subgroups.
 
== Star antiprism ==
 
{| class="wikitable"
![[List of spherical symmetry groups|Symmetry group]]
!colspan=4|Star forms
|-
!align=center| d<sub>5h</sub><BR>[2,5]<BR>(*225)
| [[Image:Pentagrammic antiprism.png|64px]]<BR>[[Pentagrammic antiprism|3.3.3.5/2]]
|-
!align=center| d<sub>5d</sub><BR>[2<sup>+</sup>,5]<BR>(2*5)
| [[Image:Pentagrammic crossed antiprism.png|64px]]<BR>[[Pentagrammic crossed-antiprism|3.3.3.5/3]]
|-
!align=center| d<sub>7h</sub><BR>[2,7]<BR>(*227)
| [[Image:Antiprism 7-2.png|64px]]<BR>[[Heptagrammic antiprism (7/2)|3.3.3.7/2]]
| [[Image:Antiprism 7-4.png|64px]]<BR>[[Heptagrammic crossed-antiprism|3.3.3.7/4]]
|-
!align=center| d<sub>7d</sub><BR>[2<sup>+</sup>,7]<BR>(2*7)
| [[Image:Antiprism 7-3.png|64px]]<BR>[[Heptagrammic antiprism (7/3)|3.3.3.7/3]]
|-
!align=center| d<sub>8d</sub><BR>[2<sup>+</sup>,8]<BR>(2*8)
| [[Image:Antiprism 8-3.png|64px]]<BR>[[Octagrammic antiprism|3.3.3.8/3]]
| [[Image:Antiprism 8-5.png|64px]]<BR>[[Octagrammic crossed-antiprism|3.3.3.8/5]]
|-
!align=center| d<sub>9h</sub><BR>[2,9]<BR>(*229)
| [[Image:Antiprism 9-2.png|64px]]<BR>[[Enneagrammic antiprism (9/2)|3.3.3.9/2]]
| [[Image:Antiprism 9-4.png|64px]]<BR>[[Enneagrammic antiprism (9/4)|3.3.3.9/4]]
|-
!align=center| d<sub>9d</sub><BR>[2<sup>+</sup>,9]<BR>(2*9)
| [[Image:Antiprism 9-5.png|64px]]<BR>[[Enneagrammic crossed-antiprism|3.3.3.9/5]]
|-
!align=center| d<sub>10d</sub><BR>[2<sup>+</sup>,10]<BR>(2*10)
| [[Image:Antiprism 10-3.png|64px]]<BR>[[Decagrammic antiprism|3.3.3.10/3]]
|-
!align=center| d<sub>11h</sub><BR>[2,11]<BR>(*2.2.11)
| [[Image:Antiprism 11-2.png|64px]]<BR>3.3.3.11/2
| [[Image:Antiprism 11-4.png|64px]]<BR>3.3.3.11/4
| [[Image:Antiprism 11-6.png|64px]]<BR>3.3.3.11/6
|-
!align=center| d<sub>11d</sub><BR>[2<sup>+</sup>,11]<BR>(2*11)
| [[Image:Antiprism 11-3.png|64px]]<BR>3.3.3.11/3
| [[Image:Antiprism 11-5.png|64px]]<BR>3.3.3.11/5
| [[Image:Antiprism 11-7.png|64px]]<BR>3.3.3.11/7
|-
!align=center| d<sub>12d</sub><BR>[2<sup>+</sup>,12]<BR>(2*12)
| [[Image:Antiprism 12-5.png|64px]]<BR>[[Dodecagrammic antiprism|3.3.3.12/5]]
| [[Image:Antiprism 12-7.png|64px]]<BR>[[Dodecagrammic crossed-antiprism|3.3.3.12/7]]
|-
| ...
|}
 
*[[Prism (geometry)|Prism]]
*[[Apeirogonal antiprism]]
*[[Grand antiprism]] – a four dimensional polytope
*[[One World Trade Center]], a building consisting primarily of an elongated square antiprism<ref name=Kabai2013>{{cite web|last=Kabai|first=Sándor|title=One World Trade Center Antiprism|url=http://demonstrations.wolfram.com/OneWorldTradeCenterAntiprism/|publisher=Wolfram Demonstrations Project|accessdate=8 October 2013}}</ref>
 
== References==
* {{cite book | author= Anthony Pugh | year= 1976 | title= Polyhedra: A visual approach | publisher= University of California Press Berkeley | location= California | isbn= 0-520-03056-7  }} Chapter 2: Archimedean polyhedra, prisma and antiprisms
{{reflist}}
 
==External links==
{{Commonscat|Antiprisms}}
*{{MathWorld |urlname=Antiprism |title=Antiprism}}
*{{GlossaryForHyperspace |anchor=Antiprism |title=Antiprism}}
**{{GlossaryForHyperspace |anchor=Prismatic |title=Prismatic polytopes}}
*[http://home.comcast.net/~tpgettys/nonconvexprisms.html Nonconvex Prisms and Antiprisms]
*[http://www.software3d.com/Prisms.php Paper models of prisms and antiprisms]
 
{{Polyhedron navigator}}
 
[[Category:Uniform polyhedra]]
[[Category:Prismatoid polyhedra]]

Revision as of 01:21, 4 March 2014

When you were younger, maybe we were one of those lucky persons whom didnt need fat reduction help. Maybe we were chasing kids, functioning at an outside job, cooking, cleaning plus living an active lifestyle. Its not that guys over 50 plus post-menopausal females are not active. But often they are less active. This can be as a result of bodily limitations, such as arthritis, or considering they have really gotten used to a more sedentary life-style in retirement or even when they are nevertheless functioning.

No matter what you eat, usually observe the part sizes. There is too much of the advantageous thing! Watch out for cream based soups and sauces; instead try tomato-based because a healthy alternative. Fried foods are a big no-no, have the fish and meats baked or grilled! Instead of a baked dessert, have fresh fruit. As another good direction, utilize cooking spray rather of oils.

I would greatly endorse a body fat check for anyone who wishes to lose fat (fat weight), whom desires to become on top form, or only desires to sustain an already healthy lifestyle. These are typically low-priced, they provide a starting point for your goals, and they provide wise encouragement to keep we on mark. But never, I reiterate never be preoccupied by the readings. Simply keep in your mind which what we see following taking a measuring is an instant inside time. The accurateness originates from we plus what we realize on top of several weeks plus months. The body fat check is a smart computer, nevertheless like any computer, it will only return to you what you place into it. Work out plus dine nourishing - the check is only a device to do the goals inside life.

The BMI index is broken down into four principal categories. Each category is represents a amount of morbidity and mortality data or wellness risk. If the bmi calculator reads lower than 18.5 you're considered underweight, when it reads between 18.5 to 25 we are at an optimal weight; when the bmi calculator reads 25 to 30 a are overweight, if it reads 30 or above than you're considered overweight (over 40 morbidly obese). Patients which score above a 30 are a major concern of the health profession. Studies have shown which being obese could have an impact on a health. These same research show how weight is shortening the average life span.

Burnello and Beatice D'Hombres of the Euopean Commision , Joint Research Center inside Ispra, Italy utilized information within the European Community House Panel, which is a compative dataset, in the study published in The Journal Economics and Human Biology.

The simplest means to keep track of the calorie and fat consumption for the day is to keep a log! Hide a small notepad and pen inside the purse particularly for this and take it where we go. Write down what you eat for the day. A superb, free site to calculate how several calories are in your food is a part of iGoogle!

If BMI is not the proper tool, what exactly is? Well, to actually accurately tell what % of the fat is fat, we have to be weighed underwater. That's a hassle, not the type of thing the average individual will do, not regularly. But, with just a tape measure, you are able to calculate a waist-hip ratio. Next we can utilize http://www.healthcalculators.org/calculators/waist_hip.asp to find when it's inside a healthy range. This really is a more sensible measure.