Minersphere::Index::Moerae

Note: since this page was published on tau day (2025-06-28), everything will be written in terms of tau instead of pi =)

The angle is a measurement of part of a circle's perimeter divided by its radius. A common and "mathematically nice" unit of angle is the radian, symbol rad, with 1 radian covering a perimeter of 1 on the unit (radius = 1) circle. The problem with it is that it covers an irrational portion of the circle, with a full turn being τ radians. The most common unit though, the degree, symbol °, fixes this issue: a full circle is 360°, a superior highly composite number.

However, coverings of a sphere are quite a bit more complicated. Unlike ordinary angles, solid angles can no longer be thought of as turns, so they don't have much use outside parts of a sphere's area. The main unit of solid angle is the steradian, symbol sr, analogous to the radian in the fact that 1 steradian covers an area of 1, and also takes up an irrational portion of the sphere, with the full sphere being 2τ sr. There is also a unit analogous to a degree, the square degree, symbol deg², equivalent to the solid angle covered by a "square" with side lengths 1°. Unlike its lower dimensional equivalent however, it covers an irrational portion of the sphere, with a full sphere having 259200τ ≈ 41252.96 deg². This is obtained by taking the sphere's area in steradians (2τ) and multiplying it by the number of degrees in a radian (360τ) squared. The general formula for any dimension n is the surface area of the unit (n−1)-sphere times 360τ to the power of n−1.

This problem persists even in higher dimensions: the glome has 23328000τ ≈ 3712766.51 cubic degrees (deg³), the phennion has 11197440000τ² ≈ 283634468.64 quartic degrees (deg⁴), and so on.

Moerae

The n-moera (pronounced /ˈmiːrə/ MEE-ruh, plural moerae /ˈmiːriː/ MEE-ree) is a multidimensional unit of n-angle, which is meant to address the problems described in the previous section. It is defined with the dimension's spherical symmetry groups, along with the previous moera unit.

Moerae are notated with a specific numeral. This numeral describes the dimensions of the surface it is in, meaning that the numeral is always one less than the dimension of the space. For example, the 3D moera unit is the dimoera (2-moera), as the sphere has a 2D surface. This numeral offset is also seen in the generic notation for higher dimensional spheres: the nD sphere is called the (n−1)-sphere for the same reason. Their names are composed of the Greek prefix representing the numeral + moera, e.g, the 3-moera used in 4D can be called the trimoera.

The name comes from Ancient Greek μοῖρα moîra, meaning degree. ~~The Greek word for moera is γράδος grádos, from Latin gradus, also meaning degree.~~

Definition

Every dimension has a "mu number", with mr_n = n-sphere areaμ_n. By definition, μ₁ = 360, making mr₁ = 1°. The next mu number μ_n+1 is then the least common multiple of μ_n and the order of that dimension's "main" spherical symmetries, those being A_k×2 = 2(k+1)!, B_k = k!2^k, k being the dimension μ_n+2, with the dimension-specific ones being H₃ = 120 (3D), F₄×2 = 2304, H₄ = 14400 (4D), E₆×2 = 103680 (6D), E₇ = 2903040 (7D) and E₈ = 696729600 (8D).

Examples

There are quite a lot less good/intuitive examples of solid angles than simple angles. The 3D moera unit is the dimoera (mr₂), of which a sphere has μ₂ = 720. It can be thought of as the area of a disc with radius arccos359360 ≈ 4.271564438°, about the angular diameter of your index finger with your arm stretched. In the sky, the (average) solid angle of the Sun or the Moon from the Earth is around 0.0038 mr₂, the solid angle of the Earth from the Moon is around 0.2 mr₂ and the solid angle of Proxima Centauri from Proxima Centauri b is around 0.036 mr₂. On the ground, the area of Russia in relation to the Earth's is 24.1 mr₂, the area of the United States is 13.9 mr₂, the area of France is 0.779 mr₂.

The disc-sphere relation is also present in higher dimensions: in 4D, the trimoera (mr₃) is the spatial angle of a ball with radius ≈2.487616488° (no closed form for above 3D), for 5D the tetramoera (mr₄) is the 4-angle of a gongyl with radius ≈5.624909215° and for 6D the pentamoera (mr₅) is the 5-angle of a phenind with radius ≈3.019278084°.

Tables

Values by dimension

Dimension	Symbol	Name	μ_n−1	In (n−1)-radians	In (n−1)-degrees
2	mr₁	Monomoera	360	τ360 ≈ 0.01745329252	1
3	mr₂	Dimoera	720	τ360 ≈ 0.01745329252	360τ ≈ 57.29577951
4	mr₃	Trimoera	57600	τ²115200 ≈ 0.0003426945973	405τ ≈ 64.45775195
5	mr₄	Tetramoera	57600	τ²86400 ≈ 0.0004569261297	194400τ² ≈ 4924.209525
6	mr₅	Pentamoera	14515200	τ³116121600 ≈ 2.136124661×10⁻⁶	3645007τ² ≈ 1318.984694
7	mr₆	Hexamoera	29030400	τ³217728000 ≈ 1.139266485×10⁻⁶	699840007τ³ ≈ 40305.20331
8	mr₇	Heptamoera	1393459200	τ⁴66886041600 ≈ 2.330150535×10⁻⁸	820125007τ³ ≈ 47232.66013
9	mr₈	Octamoera	2786918400	τ⁴146313216000 ≈ 1.065211673×10⁻⁸	9447840000049τ⁴ ≈ 1.237134666×10⁶
10	mr₉	Enneamoera	122624409600	τ⁵47087773286400 ≈ 2.079654490×10⁻¹⁰	16607531250077τ⁴ ≈ 1.383868678×10⁶
11	mr₁₀	Decamoera	245248819200	τ⁵115880067072000 ≈ 8.450659514×10⁻¹¹	170061120000000539τ⁵ ≈ 3.221936138×10⁷
12	mr₁₁	Hendecamoera	25505877196800	τ⁶97942568435712000 ≈ 6.282141603×10⁻¹³	1345210031250001001τ⁵ ≈ 1.372324061×10⁷

Symmetry group angles

Equal to the mu number μ_n−1 divided by the group order.

Dimension	A_n	B_n	F₄	H_n	G₂	E_n
2	60	45		36	30
3	30	15		6
4	480	150	50	4
5	80	50
6	2880	315				280
7	720	45				10
8	3840	135				2
9	768	15
10	3072	33
11	512	3
12	4096	13

Visual size

Dimension	(n−1)-cubic edge length	(n−1)-ball radius
2	≈0.017453292519943 rad 1°	≈0.008726646259972 rad 0.5°
3	≈0.132110909920200 rad ≈7.569397566060480°	≈0.074552863653361 rad ≈4.271564437951843°
4	≈0.069979218134216 rad ≈4.009513852715927°	≈0.043417098239037 rad ≈2.487616487801715°
5	≈0.146204755403844 rad ≈8.376915429382754°	≈0.098173185935757 rad ≈5.624909215471961°
6	≈0.073438757210399 rad ≈4.207730840841813°	≈0.052696343603348 rad ≈3.019278084243073°
7	≈0.102196860314430 rad ≈5.855448775504882°	≈0.077772871758120 rad ≈4.456057312352465°
8	≈0.081213136342975 rad ≈4.653169953472980°	≈0.065086462148078 rad ≈3.729179584522835°
9	≈0.100792795282344 rad ≈5.775001775004446°	≈0.084672862466047 rad ≈4.851397658596196°
10	≈0.083988774248796 rad ≈4.812202290933058°	≈0.073606406302395 rad ≈4.217336426252401°
11	≈0.098330684047928 rad ≈5.633933192580630°	≈0.089633102989491 rad ≈5.135598505959304°
12	≈0.077756564781601 rad ≈4.455122991421340°	≈0.073456120031304 rad ≈4.208725657200129°

Code

Most of these calculations were done with PARI/GP, a mathematics library and command line shell. Some relevant code includes:

mu = [360, 720, 57600, 57600, 14515200, 29030400, 1393459200, 2786918400, 122624409600, 245248819200, 25505877196800]
newmu(n) = lcm([mu[n-1], 2^(n+1)*(n+1)!, 2*(n+2)!])
VS(n) = { if(n % 2 == 0, pi^(n/2)/(n/2)!, bestappr(sqrt(Pi)/gamma(n/2 + 1))*pi^floor(n/2)) }
VS_Pi(n) = Pi^(n/2)/gamma(n/2+1)
totaldeg(n) = VS(n) * n * (180/pi)^(n-1)
totaldeg_Pi(n) = VS_Pi(n) * n * (180/Pi)^(n-1)
muinrad(n) = VS(n+1) * (n+1) / mu[n]
muinrad_Pi(n) = VS_Pi(n+1) * (n+1) / mu[n]
muindeg(n) = totaldeg(n+1) / mu[n]
muindeg_Pi(n) = totaldeg_Pi(n+1) / mu[n]
VS_sph(n, r) = 2*Pi^(n/2)/gamma(n/2) * intnum(x = 0,r, sin(x)^(n-1))
AS_sph(n,r) = 2*Pi^(n/2)/gamma(n/2) * sin(r)^(n-1)
inv_VS_sph(n, V) = { g = 1; for(i = 0, 25, g = (V - VS_sph(n, g))/AS_sph(n, g) + g); g }

The n-ball in n-sphere calculations were derived from this equation based on 1Rock's answer to a similar question on Mathematics Stack Exchange:

V_{n} (r) = \frac{2^{1 - \frac{n}{2}} τ^{\frac{n}{2}}}{(\frac{n}{2} - 1)!} \int_{0}^{r} {(\sin t)}^{n - 1} d t