Zero-forcing precoding: Difference between revisions

Revision as of 00:58, 8 November 2013

The golden rhombus.

In geometry, a golden rhombus is a rhombus whose diagonals are in the ratio $\frac{p}{q} = φ$ , where $φ$ is the golden ratio.

Golden rhombohedra

There are 2 convex golden rhombohedra, one constructed from six golden rhombi as a trigonal trapezohedron, a cube that has been stretched along one of its diagonal axes. This is also called the acute golden rhombohedron.

The other is the rhombic triacontahedron, constructed with 30 golden rhombic faces, alternating 3 and 5 around every vertex. The dihedral angle between adjacent rhombi of the rhombic triacontahedron is 144°, which can be constructed by placing the short sides of two golden triangles back-to-back.

File:Rhombictriacontahedron.svg

The nonconvex rhombic hexecontahedron can be constructed by 20 acute golden rhombohedron. It also represents a stellation of the rhombic triacontahedron.

Element

The internal angles of the rhombus are

2 \arctan \frac{1}{φ} \approx 63.43495

degrees

2 \arctan φ \approx 116.56505

degrees

The edge length of the golden rhombus with short diagonal $q = 1$ is

\begin{array}{rcl} e & = & \frac{1}{2} \sqrt{p^{2} + q^{2}} \\ = & \frac{1}{2} \sqrt{1 + φ^{2}} \\ = & \frac{1}{4} \sqrt{10 + 2 \sqrt{5}} \\ \approx & 0.95106 \end{array}

A golden rhombus with unit edge length has diagonal lengths

\begin{array}{rcl} p & = & \frac{φ}{e} \\ = & 2 \frac{1 + \sqrt{5}}{\sqrt{10 + 2 \sqrt{5}}} \\ \approx & 1.70130 \end{array}

\begin{array}{rcl} q & = & \frac{1}{e} \\ = & 4 \frac{1}{\sqrt{10 + 2 \sqrt{5}}} \\ \approx & 1.05146 \end{array}

References

M. Livio, The Golden Ratio: The Story of Phi, the World's Most Astonishing Number, New York: Broadway Books, p. 206, 2002.

External links

22 year-old Systems Analyst Rave from Merrickville-Wolford, has lots of hobbies and interests including quick cars, property developers in singapore and baking. Always loves visiting spots like Historic Monuments Zone of Querétaro.

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22 year-old Systems Analyst Rave from Merrickville-Wolford, has lots of hobbies and interests including quick cars, property developers in singapore and baking. Always loves visiting spots like Historic Monuments Zone of Querétaro.

Here is my web site - cottagehillchurch.com

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+[[Image:GoldenRhombus.svg|240px|thumb|The golden rhombus.]]
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+In [[geometry]], a '''golden rhombus''' is a [[rhombus]] whose diagonals are in the ratio <math>\frac{p}{q}=\varphi\!</math>, where <math>\varphi\!</math> is the [[golden ratio]].
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+== Golden rhombohedra ==
+There are 2 convex '''golden rhombohedra''', one constructed from six golden rhombi as a [[trigonal trapezohedron]], a [[cube]] that has been stretched along one of its diagonal axes. This is also called the ''acute golden rhombohedron''.
+The other is the [[rhombic triacontahedron]], constructed with 30 golden rhombic faces, alternating 3 and 5 around every vertex. The [[dihedral angle]] between adjacent rhombi of the rhombic triacontahedron is 144°, which can be constructed by placing the short sides of two [[golden triangle (mathematics)|golden triangles]] back-to-back.
+:[[File:Rhombictriacontahedron.svg|120px|]]
+The nonconvex [[rhombic hexecontahedron]] can be constructed by 20 ''acute golden rhombohedron''. It also represents a [[stellation]] of the [[rhombic triacontahedron]].
+== Element ==
+The internal angles of the rhombus are
+:<math>2\arctan\frac{1}{\varphi}\approx63.43495</math> degrees
+:<math>2\arctan\varphi\approx116.56505</math> degrees
+The edge length of the golden rhombus with short diagonal <math>q=1</math> is
+:<math>\begin{array}{rcl}e&=&\tfrac{1}{2}\sqrt{p^2+q^2}\\&=&\tfrac{1}{2}\sqrt{1+\varphi^2}\\&=&\tfrac{1}{4}\sqrt{10+2\sqrt{5}}\\&\approx&0.95106\end{array}</math>
+A golden rhombus with unit edge length has diagonal lengths
+:<math>\begin{array}{rcl}p&=&\frac{\varphi}{e}\\&=&2\frac{1+\sqrt{5}}{\sqrt{10+2\sqrt{5}}}\\&\approx&1.70130\end{array}</math>
+:<math>\begin{array}{rcl}q&=&\frac{1}{e}\\&=&4\frac{1}{\sqrt{10+2\sqrt{5}}}\\&\approx&1.05146\end{array}</math>
+==See also==
+* [[Golden rectangle]]
+* [[Golden triangle (mathematics)|Golden triangle]]
+== References==
+* M. Livio, ''The Golden Ratio: The Story of Phi, the World's Most Astonishing Number'', New York: Broadway Books, p. 206, 2002.
+==External links==
+* {{mathworld| urlname = GoldenRhombus|title= Golden Rhombus}}
+* {{mathworld| urlname = GoldenRhombohedron|title= Golden Rhombohedron}}
+[[Category:Quadrilaterals]]
+[[Category:Golden ratio]]

Zero-forcing precoding: Difference between revisions

Revision as of 00:58, 8 November 2013

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Golden rhombohedra

Element

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External links

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Zero-forcing precoding: Difference between revisions

Revision as of 00:58, 8 November 2013

Golden rhombohedra

Element

See also

References

External links

Navigation menu

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