const constants = require('@runkit/apatryda/constants/1.0.1');
const katex = require('@runkit/apatryda/katex/1.0.0');
const _ = require('lodash');
const { φ } = constants;
class RegularDodecahedron {
constructor(scale = 1, flat = false, mirror = false) {
Object.assign(this, {
flat,
mirror,
scale,
});
this.initVertices();
}
initVertices() {
this.vertices = [
..._
.times(2, i => Math.pow(-1, i) * this.scale)
.flatMap(x => _
.times(2, j => Math.pow(-1, j) * this.scale)
.map(y => [x, y])
.flatMap(xy => this.flat
? [xy]
: _
.times(2, k => Math.pow(-1, k) * this.scale)
.map(z => [...xy, z]),
)
)
,
..._
.times(2, j => Math.pow(-1, j) * φ * this.scale)
.map(y => [0, y])
.flatMap(xy => this.flat
? [xy]
: _
.times(2, k => Math.pow(-1, k) * (φ - 1) * this.scale)
.map(z => [...xy, z]),
)
,
..._
.times(2, i => Math.pow(-1, i) * (φ - 1) * this.scale)
.map(x => [x, 0])
.flatMap(xy => this.flat
? [xy]
: _
.times(2, k => Math.pow(-1, k) * φ * this.scale)
.map(z => [...xy, z]),
)
,
..._
.times(2, i => Math.pow(-1, i) * φ * this.scale)
.flatMap(x => _
.times(2, j => Math.pow(-1, j) * (φ - 1) * this.scale)
.map(y => [x, y])
.map(xy => this.flat
? xy
: [...xy, 0]
)
)
,
].map(([x, y, ...rest]) => this.mirror
? [y, x, ...rest]
: [x, y, ...rest]
);
}
}
katex`
<katex>\tan{\gamma \over 2} = \phi</katex>
<katex>\tan \alpha = {\phi \over \phi^{-1}} = \phi^2</katex>
<katex>
\alpha + \beta + {\gamma \over 2} + 2 \cdot {\pi \over 2} = 2 \pi \rArr
</katex>
<katex>
\alpha + \beta + {\gamma \over 2} = \pi \rArr
</katex>
<katex>
\beta = \pi - (\alpha + {\gamma \over 2}) \rArr
</katex>
<katex>
\tan \beta
= \tan(\pi - (\alpha + {\gamma \over 2}))
= -\tan(\alpha + {\gamma \over 2})
= -{\tan \alpha + \tan{\gamma \over 2} \over 1 - \tan \alpha \cdot \tan{\gamma \over 2}}
= {\tan \alpha + \tan{\gamma \over 2} \over \tan \alpha \cdot \tan{\gamma \over 2} - 1}
= {\phi^2 + \phi \over \phi^2 \cdot \phi - 1}
= {\phi^3 \over \phi^3 - 1}
= {\phi^3 \over (2 \phi + 1) - 1}
= {\phi^3 \over 2 \phi}
= {\phi^2 \over 2}
</katex>
<katex>
\tan \gamma
= \tan({\gamma \over 2} + {\gamma \over 2})
= {2 \tan{\gamma \over 2} \over 1 - \tan^2{\gamma \over 2}}
= {2 \phi \over 1 - \phi^2}
= {2 \phi \over -\phi}
= -2
</katex>
<katex>
\sin \alpha
= \sqrt{\tan^2 \alpha \over 1 + \tan^2 \alpha}
= \sqrt{\phi^4 \over 1 + \phi^4}
= \sqrt{3 \phi + 2 \over 3 \phi + 3}
= \sqrt{(6 - 9)\phi + (9 - 6 - 6) \over 9 - 9 - 9}
= \sqrt{3 \phi + 3 \over 9}
= {\sqrt{3 \phi^2} \over 3}
= {\sqrt{3} \over 3} \phi
</katex>
<katex>
\cos \alpha
= \sqrt{1 \over 1 + \tan^2 \alpha}
= \sqrt{1 \over 1 + \phi^4}
= \sqrt{1 \over 3 \phi + 3}
= \sqrt{(3 - 0)\phi + (0 - 3 - 3) \over 9 - 9 - 9}
= \sqrt{6 - 3 \phi \over 9}
= {\sqrt{3(2 - \phi)} \over 3}
= {\sqrt{3 \over \phi^2} \over 3}
= {\sqrt{3} \over 3 \phi}
= {\sqrt{3} \over 3} (\phi - 1)
</katex>
<katex>
\sin \beta
= \sqrt{\tan^2 \beta \over 1 + \tan^2 \beta}
= \sqrt{\phi^4 \over 4 + \phi^4}
= \sqrt{3 \phi + 2 \over 3 \phi + 6}
= \sqrt{(6 - 18)\phi + (9 - 6 - 12) \over 9 - 18 - 36}
= \sqrt{12 \phi + 9 \over 45}
= \sqrt{4 \phi + 3 \over 15}
</katex>
<katex>
\cos \beta
= \sqrt{1 \over 1 + \tan^2 \beta}
= \sqrt{4 \over 4 + \phi^4}
= \sqrt{4 \over 3 \phi + 6}
= \sqrt{(12 - 0)\phi + (0 - 12 - 24) \over 9 - 18 - 36}
= \sqrt{36 - 12 \phi \over 45}
= 2 \sqrt{3 - \phi \over 15}
</katex>
<katex>
\sin{\gamma \over 2}
= \sqrt{\tan^2{\gamma \over 2} \over 1 + \tan^2{\gamma \over 2}}
= \sqrt{\phi^2 \over 1 + \phi^2}
= \sqrt{\phi + 1 \over \phi + 2}
= \sqrt{(1 - 2)\phi + (1 - 1 - 2) \over 1 - 2 - 4}
= \sqrt{\phi + 2 \over 5}
</katex>
<katex>
\cos{\gamma \over 2}
= \sqrt{1 \over 1 + \tan^2{\gamma \over 2}}
= \sqrt{1 \over 1 + \phi^2}
= \sqrt{1 \over \phi + 2}
= \sqrt{(1 - 0)\phi + (0 - 1 - 2) \over 1 - 2 - 4}
= \sqrt{3 - \phi \over 5}
</katex>
<katex>
\sin \gamma
= 2 \sin{\gamma \over 2} \cos{\gamma \over 2}
= 2 \cdot \sqrt{\phi + 2 \over 5} \cdot \sqrt{3 - \phi \over 5}
= {2 \over 5} \sqrt{(\phi + 2)(3 - \phi)}
= {2 \over 5} \sqrt{3 \phi - \phi^2 + 6 - 2 \phi}
= {2 \over 5} \sqrt{3 \phi - \phi - 1 + 6 - 2 \phi}
= {2 \sqrt{5} \over 5}
= {2 \cdot (2 \phi - 1) \over 5}
= {4 \phi - 2 \over 5}
</katex>
<katex>
\cos \gamma
= \cos^2{\gamma \over 2} - \sin^2{\gamma \over 2}
= ({3 - \phi \over 5}) - ({\phi + 2 \over 5})
= {3 - \phi - \phi - 2 \over 5}
= {1 - 2 \phi \over 5}
= -{\sqrt{5} \over 5}
</katex>
<katex>\alpha = \arctan \phi^2</katex>
<katex>\beta = \arctan {\phi^2 \over 2}</katex>
<katex>\gamma = 2 \arctan \phi</katex>
`;
Object.assign(RegularDodecahedron, (() => {
const tan = {
alpha: φ + 1,
beta: 0.5 * φ + 0.5,
gamma: -2,
half: {
gamma: φ,
},
};
const alpha = Math.atan(tan.alpha);
const beta = Math.atan(tan.beta);
const half = {
gamma: Math.atan(tan.half.gamma),
};
const gamma = 2 * half.gamma;
const sin = {
alpha: φ / Math.sqrt(3),
beta: Math.sqrt((4 * φ + 3) / 15),
gamma: (4 * φ - 2) / 5,
half: {
gamma: Math.sqrt((φ + 2) / 5),
},
};
const cos = {
alpha: (φ - 1) / Math.sqrt(3),
beta: 2 * Math.sqrt((3 - φ) / 15),
gamma: (1 - 2 * φ) / 5,
half: {
gamma: Math.sqrt((3 - φ) / 5),
},
};
return {
alpha,
beta,
gamma,
half,
cos,
sin,
tan,
};
})());
const regularDodecahedron = new RegularDodecahedron();
function testValues(o, fn, p = []) {
return _.mapValues(o, (v, k) => _.isPlainObject(v)
? testValues(v, fn, [...p, k])
: Math.log(Math.abs(v - fn(_.get(RegularDodecahedron, [...p, k])))) / Math.LN2
);
}
testValues(RegularDodecahedron.sin, Math.sin);
testValues(RegularDodecahedron.cos, Math.cos);
testValues(RegularDodecahedron.tan, Math.tan);
module.exports = {
RegularDodecahedron,
regularDodecahedron,
};
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