Snub cube's angles
I am trying to build a snub cube. I have made $6$ squares and $32$ equilateral triangles (out of perler beads if you're curious).
I am trying to figure out the angles at which I adjoin the squares to the triangles, and the triangles to other triangles.
I have found a few formulas, but I think I am a bit overwhelmed by the vocabulary used and do not understand what the listed variables are.
H. Rajpoot says "There is a general expression of the solid angle subtended by the snub cube at any of its $24$ vertices is given by the general expression
begin{align}Omega&=2sin^{-1}left(frac{(1-sqrt{1-K^2})-sqrt{2K^2-1}}{K^2sqrt{2}}right)+8sin^{-1}left(frac{(1-sqrt{1-K^2})-sqrt{4K^2-1}}{2K^2sqrt{3}}right)\&approx 3.589629551 space sr,end{align}
where $Kapprox 0.928191378"$.
and Felix Marin says that the formula to find the angles is $$
cosleft(vphantom{Large A}angle{rm ABC}right)
=
{left(vec{A} - vec{B}right)cdotleft(vec{C} - vec{B}right)
over
leftvertvec{A} - vec{B}rightvert;leftvertvec{C} - vec{B}rightvert}
$$ where $A$, $B$, and $C$ are are vectors $A:[x_1,y_1,z_1]$, $B:[x_2,y_2,z_2]$, and $C:[x_3,y_3,z_3]$.
I suppose, I am completely overwhelmed.
I have a sight feeling that finding the 'subtended angle' is not the same as the angle I am trying to find. Is that true?
What is $s$? $r$?
Why are $A$, $B$, & $C$ vectors and how do I know which vectors to use?
I saw online, here that the the coordinates for the vertices of a snub cube are all the even permutations of $(±1, ±1/t, ±t)$ with an even number of plus signs, along with all the odd permutations with an odd number of plus signs, where $t ≈ 1.83929$ is the tribonacci constant.
Are these the values I am supposed to use to find the vectors to use the second equation?
Is there an easier way to do this? I fell like I have way over-complicated this.
edit: okay, I found this website that says the square-triangle angle is $142$ degrees, $59$ minutes and the triangle-triangle angle is $153$ degrees, $14$ minutes. Would still be stoked to know how on earth to figure this out on my own. thanks!
geometry trigonometry vectors euclidean-geometry solid-geometry
asked Dec 12 '18 at 22:14
nessness
375
add a comment |
I am trying to build a snub cube. I have made $6$ squares and $32$ equilateral triangles (out of perler beads if you're curious).
I am trying to figure out the angles at which I adjoin the squares to the triangles, and the triangles to other triangles.
I have found a few formulas, but I think I am a bit overwhelmed by the vocabulary used and do not understand what the listed variables are.
H. Rajpoot says "There is a general expression of the solid angle subtended by the snub cube at any of its $24$ vertices is given by the general expression
begin{align}Omega&=2sin^{-1}left(frac{(1-sqrt{1-K^2})-sqrt{2K^2-1}}{K^2sqrt{2}}right)+8sin^{-1}left(frac{(1-sqrt{1-K^2})-sqrt{4K^2-1}}{2K^2sqrt{3}}right)\&approx 3.589629551 space sr,end{align}
where $Kapprox 0.928191378"$.
and Felix Marin says that the formula to find the angles is $$
cosleft(vphantom{Large A}angle{rm ABC}right)
=
{left(vec{A} - vec{B}right)cdotleft(vec{C} - vec{B}right)
over
leftvertvec{A} - vec{B}rightvert;leftvertvec{C} - vec{B}rightvert}
$$ where $A$, $B$, and $C$ are are vectors $A:[x_1,y_1,z_1]$, $B:[x_2,y_2,z_2]$, and $C:[x_3,y_3,z_3]$.
I suppose, I am completely overwhelmed.
I have a sight feeling that finding the 'subtended angle' is not the same as the angle I am trying to find. Is that true?
What is $s$? $r$?
Why are $A$, $B$, & $C$ vectors and how do I know which vectors to use?
I saw online, here that the the coordinates for the vertices of a snub cube are all the even permutations of $(±1, ±1/t, ±t)$ with an even number of plus signs, along with all the odd permutations with an odd number of plus signs, where $t ≈ 1.83929$ is the tribonacci constant.
Are these the values I am supposed to use to find the vectors to use the second equation?
Is there an easier way to do this? I fell like I have way over-complicated this.
edit: okay, I found this website that says the square-triangle angle is $142$ degrees, $59$ minutes and the triangle-triangle angle is $153$ degrees, $14$ minutes. Would still be stoked to know how on earth to figure this out on my own. thanks!
geometry trigonometry vectors euclidean-geometry solid-geometry
asked Dec 12 '18 at 22:14
nessness
375
2
I think $sr$ is the unit steradian of solid angles (i.e., $sr$ is not a product of $s$ and $r$).
– user614671
Dec 13 '18 at 23:39
add a comment |
I am trying to build a snub cube. I have made $6$ squares and $32$ equilateral triangles (out of perler beads if you're curious).
I am trying to figure out the angles at which I adjoin the squares to the triangles, and the triangles to other triangles.
I have found a few formulas, but I think I am a bit overwhelmed by the vocabulary used and do not understand what the listed variables are.
H. Rajpoot says "There is a general expression of the solid angle subtended by the snub cube at any of its $24$ vertices is given by the general expression
begin{align}Omega&=2sin^{-1}left(frac{(1-sqrt{1-K^2})-sqrt{2K^2-1}}{K^2sqrt{2}}right)+8sin^{-1}left(frac{(1-sqrt{1-K^2})-sqrt{4K^2-1}}{2K^2sqrt{3}}right)\&approx 3.589629551 space sr,end{align}
where $Kapprox 0.928191378"$.
and Felix Marin says that the formula to find the angles is $$
cosleft(vphantom{Large A}angle{rm ABC}right)
=
{left(vec{A} - vec{B}right)cdotleft(vec{C} - vec{B}right)
over
leftvertvec{A} - vec{B}rightvert;leftvertvec{C} - vec{B}rightvert}
$$ where $A$, $B$, and $C$ are are vectors $A:[x_1,y_1,z_1]$, $B:[x_2,y_2,z_2]$, and $C:[x_3,y_3,z_3]$.
I suppose, I am completely overwhelmed.
I have a sight feeling that finding the 'subtended angle' is not the same as the angle I am trying to find. Is that true?
What is $s$? $r$?
Why are $A$, $B$, & $C$ vectors and how do I know which vectors to use?
I saw online, here that the the coordinates for the vertices of a snub cube are all the even permutations of $(±1, ±1/t, ±t)$ with an even number of plus signs, along with all the odd permutations with an odd number of plus signs, where $t ≈ 1.83929$ is the tribonacci constant.
Are these the values I am supposed to use to find the vectors to use the second equation?
Is there an easier way to do this? I fell like I have way over-complicated this.
edit: okay, I found this website that says the square-triangle angle is $142$ degrees, $59$ minutes and the triangle-triangle angle is $153$ degrees, $14$ minutes. Would still be stoked to know how on earth to figure this out on my own. thanks!
geometry trigonometry vectors euclidean-geometry solid-geometry
asked Dec 12 '18 at 22:14
nessness
375
I am trying to build a snub cube. I have made $6$ squares and $32$ equilateral triangles (out of perler beads if you're curious).
I am trying to figure out the angles at which I adjoin the squares to the triangles, and the triangles to other triangles.
I have found a few formulas, but I think I am a bit overwhelmed by the vocabulary used and do not understand what the listed variables are.
H. Rajpoot says "There is a general expression of the solid angle subtended by the snub cube at any of its $24$ vertices is given by the general expression
begin{align}Omega&=2sin^{-1}left(frac{(1-sqrt{1-K^2})-sqrt{2K^2-1}}{K^2sqrt{2}}right)+8sin^{-1}left(frac{(1-sqrt{1-K^2})-sqrt{4K^2-1}}{2K^2sqrt{3}}right)\&approx 3.589629551 space sr,end{align}
where $Kapprox 0.928191378"$.
and Felix Marin says that the formula to find the angles is $$
cosleft(vphantom{Large A}angle{rm ABC}right)
=
{left(vec{A} - vec{B}right)cdotleft(vec{C} - vec{B}right)
over
leftvertvec{A} - vec{B}rightvert;leftvertvec{C} - vec{B}rightvert}
$$ where $A$, $B$, and $C$ are are vectors $A:[x_1,y_1,z_1]$, $B:[x_2,y_2,z_2]$, and $C:[x_3,y_3,z_3]$.
I suppose, I am completely overwhelmed.
I have a sight feeling that finding the 'subtended angle' is not the same as the angle I am trying to find. Is that true?
What is $s$? $r$?
Why are $A$, $B$, & $C$ vectors and how do I know which vectors to use?
I saw online, here that the the coordinates for the vertices of a snub cube are all the even permutations of $(±1, ±1/t, ±t)$ with an even number of plus signs, along with all the odd permutations with an odd number of plus signs, where $t ≈ 1.83929$ is the tribonacci constant.
Are these the values I am supposed to use to find the vectors to use the second equation?
Is there an easier way to do this? I fell like I have way over-complicated this.
edit: okay, I found this website that says the square-triangle angle is $142$ degrees, $59$ minutes and the triangle-triangle angle is $153$ degrees, $14$ minutes. Would still be stoked to know how on earth to figure this out on my own. thanks!
geometry trigonometry vectors euclidean-geometry solid-geometry
geometry trigonometry vectors euclidean-geometry solid-geometry
asked Dec 12 '18 at 22:14
nessness
375
asked Dec 12 '18 at 22:14
nessness
375
edited Dec 13 '18 at 23:47
user614671
asked Dec 12 '18 at 22:14
nessness
375
asked Dec 12 '18 at 22:14
nessness
375
asked Dec 12 '18 at 22:14
nessness
375
375
2
I think $sr$ is the unit steradian of solid angles (i.e., $sr$ is not a product of $s$ and $r$).
– user614671
Dec 13 '18 at 23:39
add a comment |
2
I think $sr$ is the unit steradian of solid angles (i.e., $sr$ is not a product of $s$ and $r$).
– user614671
Dec 13 '18 at 23:39
2
2
I think $sr$ is the unit steradian of solid angles (i.e., $sr$ is not a product of $s$ and $r$).
– user614671
Dec 13 '18 at 23:39
I think $sr$ is the unit steradian of solid angles (i.e., $sr$ is not a product of $s$ and $r$).
– user614671
Dec 13 '18 at 23:39
add a comment |
1 Answer
1
active
oldest
votes
For the mere requested values you might want to have a look here.
A more descriptive way on the derivation of these values might be found already in the old German book of Max Brückner, "Vielecke und Vielflache, Theorie und Geschichte", Leipzig, Teubner Verlag (1900), at page 139.
--- rk
answered Dec 13 '18 at 23:08
Dr. Richard KlitzingDr. Richard Klitzing
1,42616
add a comment |
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active
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For the mere requested values you might want to have a look here.
A more descriptive way on the derivation of these values might be found already in the old German book of Max Brückner, "Vielecke und Vielflache, Theorie und Geschichte", Leipzig, Teubner Verlag (1900), at page 139.
--- rk
answered Dec 13 '18 at 23:08
Dr. Richard KlitzingDr. Richard Klitzing
1,42616
add a comment |
For the mere requested values you might want to have a look here.
A more descriptive way on the derivation of these values might be found already in the old German book of Max Brückner, "Vielecke und Vielflache, Theorie und Geschichte", Leipzig, Teubner Verlag (1900), at page 139.
--- rk
answered Dec 13 '18 at 23:08
Dr. Richard KlitzingDr. Richard Klitzing
1,42616
add a comment |
For the mere requested values you might want to have a look here.
A more descriptive way on the derivation of these values might be found already in the old German book of Max Brückner, "Vielecke und Vielflache, Theorie und Geschichte", Leipzig, Teubner Verlag (1900), at page 139.
--- rk
answered Dec 13 '18 at 23:08
Dr. Richard KlitzingDr. Richard Klitzing
1,42616
For the mere requested values you might want to have a look here.
A more descriptive way on the derivation of these values might be found already in the old German book of Max Brückner, "Vielecke und Vielflache, Theorie und Geschichte", Leipzig, Teubner Verlag (1900), at page 139.
--- rk
answered Dec 13 '18 at 23:08
Dr. Richard KlitzingDr. Richard Klitzing
1,42616
answered Dec 13 '18 at 23:08
Dr. Richard KlitzingDr. Richard Klitzing
1,42616
answered Dec 13 '18 at 23:08
Dr. Richard KlitzingDr. Richard Klitzing
1,42616
answered Dec 13 '18 at 23:08
Dr. Richard KlitzingDr. Richard Klitzing
1,42616
1,42616
add a comment |
add a comment |
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2
I think $sr$ is the unit steradian of solid angles (i.e., $sr$ is not a product of $s$ and $r$).
– user614671
Dec 13 '18 at 23:39