Golden rhombus

(See the characterizations and the basic properties of the general rhombus for angle properties.)

The internal supplementary angles of the golden rhombus are:{{citation

| last = Ogawa | first = Tohru

| date = January 1987

| doi = 10.4028/www.scientific.net/msf.22-24.187

| journal = Materials Science Forum

| pages = 187–200

| title = Symmetry of three-dimensional quasicrystals

| volume = 22-24| s2cid = 137677876

}}. See in particular table 1, p. 188.

• Acute angle: $\backslash alpha=2\backslash arctan\{1\backslash over\backslash varphi\}$ ;

:by using the arctangent addition formula (see inverse trigonometric functions):

:$\backslash alpha=\backslash arctan\{\{2\backslash over\backslash varphi\}\backslash over\{1-(\{1\backslash over\backslash varphi\})^2\}\}=\backslash arctan\{\{2\backslash over\backslash varphi\}\backslash over\{1\backslash over\backslash varphi\}\}=\backslash arctan2\backslash approx63.43495^\backslash circ.$

:

• Obtuse angle: $\backslash beta=2\backslash arctan\backslash varphi=\backslash pi-\backslash arctan2\backslash approx116.56505^\backslash circ,$

:which is also the dihedral angle of the dodecahedron.{{citation

| last = Gevay | first = G.

| date = June 1993

| doi = 10.1080/01411599308210255

| issue = 1–3

| journal = Phase Transitions

| pages = 47–50

| title = Non-metallic quasicrystals: Hypothesis or reality?

| volume = 44| bibcode = 1993PhaTr..44...47G

}}

:Note: an "anecdotal" equality: $\backslash pi-$

\arctan2=\arctan1+

\arctan3~.

:

By using the parallelogram law (see the basic properties of the general rhombus):{{mathworld|id=Rhombus|title=Rhombus}}

The edge length of the golden rhombus in terms of the diagonal length $d$ is:

• $a=\{1\backslash over2\}\backslash sqrt\{d^2+(\backslash varphi\; d)^2\}=\{1\backslash over2\}\backslash sqrt\{1+\backslash varphi^2\}~d=\{\{\backslash sqrt\{2+\backslash varphi\}\}\backslash over2\}~d=\{1\backslash over4\}\backslash sqrt\{10+2\backslash sqrt5\}~d\backslash approx0.95106~d~.~$ Hence:

:

The diagonal lengths of the golden rhombus in terms of the edge length $a$ are:

• $d=\{2a\backslash over\backslash sqrt\{2+\backslash varphi\}\}=2\backslash sqrt\{\{3-\backslash varphi\}\backslash over5\}~a=\backslash sqrt\{2-\{2\backslash over\backslash sqrt5\}\}~a\backslash approx1.05146~a~.$

:

• $D=\{2\backslash varphi\; a\backslash over\backslash sqrt\{2+\backslash varphi\}\}=2\backslash sqrt\{\{2+\backslash varphi\}\backslash over5\}~a=\backslash sqrt\{2+\{2\backslash over\backslash sqrt5\}\}~a\backslash approx1.70130~a~.$

:

• By using the area formula of the general rhombus in terms of its diagonal lengths $D$ and $d$ :

:

:The area of the golden rhombus in terms of its diagonal length $d$ is:

:$A\; =\; \{\{(\backslash varphi\; d)\backslash cdot\; d\}\backslash over2\}\; =\; \{\{\backslash varphi\}\backslash over2\}~d^2\; =\; \{\{1+\backslash sqrt5\}\backslash over4\}~d^2\; \backslash approx\; 0.80902~d^2~.$

• By using the area formula of the general rhombus in terms of its edge length $a$ :

:

:The area of the golden rhombus in terms of its edge length $a$ is:{{mathworld |urlname=GoldenRhombus |title=Golden Rhombus}}

:$A\; =\; (\backslash sin(\backslash arctan2))~a^2\; =\; \{2\backslash over\backslash sqrt5\}~a^2\; \backslash approx\; 0.89443~a^2~.$

:

Note: $\backslash alpha+\backslash beta\; =\; \backslash pi$ , hence: $\backslash sin\backslash alpha\; =\; \backslash sin\backslash beta~.$

:

Several notable polyhedra have golden rhombi as their faces.

They include the two golden rhombohedra (with six faces each), the Bilinski dodecahedron (with 12 faces),

the rhombic icosahedron (with 20 faces),

the rhombic triacontahedron (with 30 faces), and

the nonconvex rhombic hexecontahedron (with 60 faces). The first five of these are the only convex polyhedra with golden rhomb faces, but there exist infinitely many nonconvex polyhedra having this shape for all of their faces.{{citation

|last = Grünbaum

|first = Branko

|authorlink = Branko Grünbaum

|doi = 10.1007/s00283-010-9138-7

|issue = 4

|journal = The Mathematical Intelligencer

|mr = 2747698

|pages = 5–15

|title = The Bilinski dodecahedron and assorted parallelohedra, zonohedra, monohedra, isozonohedra, and otherhedra

|url = https://dlib.lib.washington.edu/researchworks/bitstream/handle/1773/15593/Bilinski_dodecahedron.pdf

|volume = 32

|year = 2010

|url-status = dead

|archiveurl = https://web.archive.org/web/20150402132516/https://dlib.lib.washington.edu/researchworks/bitstream/handle/1773/15593/Bilinski_dodecahedron.pdf

|archivedate = 2015-04-02

|hdl= 1773/15593

|s2cid = 120403108

|hdl-access= free

}}.

File:Acute_golden_rhombohedron.png|Acute golden rhombohedron

File:Flat_golden_rhombohedron.png|Obtuse golden rhombohedron

File:Bilinski dodecahedron.png|Bilinski dodecahedron

File:Rhombic icosahedron.png|rhombic icosahedron

File:Rhombictriacontahedron.svg|rhombic triacontahedron

File:Rhombic hexecontahedron.png|rhombic hexecontahedron

• Golden triangle

{{reflist}}

{{Metallic ratios}}

Category:Types of quadrilaterals

Category:Golden ratio