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Internal solid angles of regular simplices
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Numbers
$d$
$k$ 
$\beta_{d,k}$
2
0:
1/6
comment: A vertex of an equilateral triangle has angle $\pi/3$, which is $1/6$ of the full circle.
2
1:
1/2
3
0:
0.04386991402295545262814166941682318348747148707955021546595372351650105650808168515265886963183088481
comment: The solid angle at a vertex of a regular tetrahedron is $4\pi\beta_{3,0}=3\arccos(1/3)-\pi$ steradians.
3
1:
0.1959132760153036350854277796112154556583143247197001436439691490110007043387211234351059130878872565
3
2:
1/2
4
0:
0.009784688372416878114524903335757872207967784376231620899860783308746406720201523423725250574067449095
4
1:
0.06467703255862531717178735500363680831195167656434743134979117496311961008030228513558787586110117364
4
2:
0.2097846883724168781145249033357578722079677843762316208998607833087464067202015234237252505740674491
4
3:
1/2
5
0:
0.001922043736984349261570562052372220975086823559286757809311492523143875125966577705926921649756909278
5
1:
0.01872170907030627389869108173619142049713684268059283557452076921683606315322891694455815027462953565
5
2:
0.07692933736326880129109428537286379816065316988531719867619425831136776965426342879328107242209015791
5
3:
0.2179528915755125341940628569152425321071021132568781324507961722075785131028422858621873816147267719
5
4:
1/2
6
0:
0.0003406951396911147500904258145159268548415934754218403041691124882731588869744941577677086335360869882
6
1:
0.004872743237305553220116280004021157796256008676598533132778385006414976264912649493202988390225409843
6
2:
0.02529890777586538394238393466633321071561694815870708462461561332456391097612708475386604058375452393
6
3:
0.08502471572141474398150613399708712139567181703210150705725638898299688070524303743487726784149653999
6
4:
0.2233498104809431626543374226647247475971145446880676713715042593219979204701620249565848452276643600
6
5:
1/2
7
0:
0.00005540935369859942446641838097847515797991674552496941008707432562143588383267668441977628447926084230
7
1:
0.001160196497658583704636236797610090195046640119284759049750817307898814629728188866641814772928711131
7
2:
0.007594271025140070868430623934429503782912486680557948530839175831881887542152000164637153905489124499
7
3:
0.03022323423856105948894183435082690020774235450649811232313357270964543563246995219579969770416176845
7
4:
0.09077828874275754671235437716558264804186603975141239936619685356533902092341372819491725421294917797
7
5:
0.2271855258285050311415695847770550986945773598342749329107979023768926806156091521299448361419661186
7
6:
1/2
8
0:
0.000008368841096666050153806261024040077706810369927154783083915297884296567795120912914146646383357132183
8
1:
0.0002559009763379599894675243626245544454611536300823117881189749923561928238003860667160864332843390182
8
2:
0.002107842331203002712113466231170765563466646479895985762111069001299892399171690189814543072820399611
8
3:
0.009918598704985813893441600265849482226862014506658090564871712726247334140121940406852203255728430661
8
4:
0.03402089564461021077115720781148268590065620107337733828182367253602592732297806863493967962998419952
8
5:
0.09508018424392382782067085155771433951486709290607115308381248116829049050039393870829819749153924088
8
6:
0.2300534561626158852137805677051428930099113952707141020558749874455269936669292924721987983276928273
8
7:
1/2
9
0:
0.000001184659403048245260021493928250488229433753389837871356353311085461133428833624109844194702716024372
9
1:
0.00005278751814745991464928578707198035258992301266935459586848232973000731496565306313024976190209802550
9
2:
0.0005462661151670698614895220594186849265004672189422634328725151212157182439385126771668183128920742702
9
3:
0.003035316984722707609845804269716706673202081393397328017498624520654750926825090564427370349275870110
9
4:
0.01187614740834568227680332856783827951844751804167947270080866139132918700328494883006153364368346108
9
5:
0.03702998616210826216548753548590342031301300463834594326572783265286095737934141759310255051079832311
9
6:
0.09841929079815498388214930608815216275845099613251123127810655214449392386704115709161690557998740801
9
7:
0.2322795271987699892547662040587681085056339974216741541854043680963292825780274380610779370533249387
9
8:
1/2
10
0:
1.583018664979023617667655384186981303406254283904610883972310824345579934064168883182097360562607594e-7
10
1:
0.00001026008265455376100917049884372727397293049556309543726610683542419492938460330123564850800464387366
10
2:
0.0001332028850936177527205942221736933384507125273273444992205685058827681626668356630426813659935106237
10
3:
0.0008729694009954713064938865713345030156034730498845937369157654245344584371296240292713536590655543397
10
4:
0.003892452522317547976008382750677273249764354006651150816028216793334853045458413159118999302472754188
10
5:
0.01352910544328957518454894644309107641991287735932142509081896289543975435893386947456665052278403326
10
6:
0.03946950196268586809461149828854089577432132510614602929271808617640539490602710272316273616920220250
10
7:
0.1010867896780550570311878795669566978095342612436261888845857515273442397436803324000041139401328838
10
8:
0.2340578597853700380207919197113044652063561741624174592563905010182294931624535549333360759600885892
10
9:
1/2
11
0:
2.008396792458658008908257709132386796136884156420920473088384677814929833495054159970487382501708939e-8
11
1:
0.000001890325150435258736380794367293649166465270111444440349652658801415058389352992745516478091791260791
11
2:
0.00003074987193848506095611004422036487143360471353383036318024716398516081036396705394237387326790378784
11
3:
0.0002374475996721987345013289280898422182596516654785422070896445411517935476207081408105872308683360338
11
4:
0.001205498589219555450591365181102895054119297440157666113379566382399459480483700671068671932840013566
11
5:
0.004667155735633114485017696811185088183489229066854631735749043213776812223957855641863624434464431572
11
6:
0.01493555296546346656737370136630990845708328202438236744127184946249529147123925696654768432806528984
11
7:
0.04148558626825726533485655547454668681808958869061279902377124735809190907190633439088357748005069738
11
8:
0.1032670476231078180618606123920340851528844138212897780708937613596406887251159474063967556232368722
11
9:
0.2355113650820718787079070749280227234352562758808598520472625075730937924834106316042645037488245814
11
10:
1/2
12
0:
2.430755715712474851574525601700902614692840094512023056923455621127895170440319446939258606543351681e-9
12
1:
3.317620826501624699811564874821752602571026216960308774943372943446112983491245147623643118793394512e-7
12
2:
0.000006754462964457422044543824228260879895533179954060185002579517889554774649308891675664059329149880458
12
3:
0.00006139796523302865169840099920326215161370905430503301374062382915480228834459630331046017731320568705
12
4:
0.0003546394752947990186707999022754099351678494646650146096360483200035715919801328540522091833506390462
12
5:
0.001528327111555218395843261283301518061823474463549489639314081136599530493436264981645620008203749044
12
6:
0.005361665693239549621850883697301371629122252667678691621518718564187744119647711675495432622585550190
12
7:
0.01614298831803446502130495205679588976040938775933693148318864711075710281814345969376317918570978966
12
8:
0.04317888464054945589076912398930825963063715355193667496765193372978414737764103133212397623379167998
12
9:
0.1050825339700035048233707147498848555897100976723028535615647123282219593755754711819280568392616462
12
10:
0.2367216893133356698822471431665899037264733984482019023743764748854813062503836474546187045595077641
12
11:
1/2
Definition
The internal solid angle $\beta_{d,k}$ at one $k$-face of a regular $d$-simplex is the Gaussian measure of its tangent cone in the simplex's direction space [1].
Parameters
$d$
—   simplex dimension ($d\ge2$)
$k$
—   face dimension ($0\le k<d$)
Formulas
(1)
$\displaystyle \beta_{d,k}=g_{d-k}\!\left(-\frac1{d+1}\right)=\frac1{\sqrt{2\pi}}\int_0^\infty e^{-x^2/2}\left[\Phi\!\left(\frac{ix}{\sqrt{d+1}}\right)^{d-k}+\Phi\!\left(-\frac{ix}{\sqrt{d+1}}\right)^{d-k}\right]\,\mathrm dx.$ This is Proposition 1.2(b) and equation (1.19) of [1], with $i^2=-1$ and positive real square roots.
(2)
$\displaystyle \beta_{d,d-1}=\frac12,\qquad \beta_{2,0}=\frac16.$ A facet has a half-space as tangent cone; an equilateral triangle has interior angle $\pi/3$ at each vertex.
(3)
$\displaystyle \beta_{d,d-2}=\frac{\arccos(1/d)}{2\pi}=\frac14-\frac1{2\pi}\arcsin\!\left(\frac1d\right).$ The pointed tangent cone is a planar wedge. The inverse trigonometric functions take their real principal values.
(4)
$\displaystyle \beta_{d,d-3}=\frac18-\frac3{4\pi}\arcsin\!\left(\frac1d\right),\qquad d\ge3.$ This is the spherical-triangle case, equivalently Proposition 1.4(f) of [1].
(5)
$\displaystyle \beta_{d,k}=\frac12-\sum_{\substack{0\le j<m\\j\equiv m\pmod2}}\binom mj\beta_{k+j,k}\gamma_{d,k+j},\qquad m=d-k.$ Here $\gamma_{d,s}$ is the external solid angle at an $s$-face; for this identity set $\beta_{a,a}=1$ and $\beta_{a+1,a}=1/2$ for $a\ge0$. The relation follows from Proposition 1.3 and the conic Gauss–Bonnet identity in Remark 1.1 of [1].
Comments
(6)
Let $S_d=\operatorname{conv}(e_0,\ldots,e_d)\subset\mathbb R^{d+1}$, regarded in its own $d$-dimensional affine hull. The full sphere has angular measure $1$. Let $V=\{v\in\mathbb R^{d+1}:\sum_i v_i=0\}$. If $x$ lies in the relative interior of a face $F$, its tangent cone is $T_F=\{v\in V:x+\varepsilon v\in S_d\text{ for some }\varepsilon>0\}$. Removing its $k$-dimensional lineality gives the same Gaussian measure in $d-k$ dimensions.
(7)
A $k$-face has $k+1$ vertices. All faces of that dimension have the same angle, so multiplication by $\binom{d+1}{k+1}$ gives the sum over them. Uniform scaling leaves the angles unchanged. For a vertex of a three-dimensional simplex, the solid angle in steradians is $4\pi\beta_{3,0}$. In [2], $\Delta_n$ has $n$ vertices and the face index counts vertices, so its indices are $n=d+1$ and $k_{\mathrm{source}}=k+1$.
(8)
The external solid angles measure the normal cones in their own linear spans. They are a distinct quantity and are not generally complementary to these internal angles.
(9)
Write $g_m(r)=\Pr(X_1<0,\ldots,X_m<0)$ for a centered Gaussian vector with covariance $I_m+r\mathbf1\mathbf1^T$, where $m\ge1$ and $r>-1/m$, and set $g_0(r)=1$. Here $I_m$ is the identity matrix and $\mathbf1$ is the vector of ones. The internal-angle covariance parameter is $r=-1/(d+1)$, and the correlation between distinct coordinates is $r/(1+r)=-1/d$. Let $\Phi(z)=(1+\operatorname{erf}(z/\sqrt2))/2$, where $\operatorname{erf}$ is the error function.
Programs
(P1)
Python
# Numerical approximation; the attached generator certifies the digits.
import mpmath as mp
from fractions import Fraction
mp.mp.dps = 125

def beta(d, k):
    if not (d >= 2 and 0 <= k < d):
        raise ValueError('Require d >= 2 and 0 <= k < d')
    if k == d - 1:
        return Fraction(1, 2)
    if (d, k) == (2, 0):
        return Fraction(1, 6)
    scale = mp.sqrt(2*(d+1))
    cutoff = mp.sqrt(mp.mpf(6)*(d+1)*(mp.mp.dps+30)/(k+1))
    def integrand(x):
        p = (1 + mp.erf(1j*x/scale))/2
        return mp.exp(-x*x/2)*mp.re(p**(d-k))
    return mp.sqrt(2/mp.pi)*mp.quad(integrand, [0, 2, 4, 8, 16, cutoff])

print(mp.nstr(beta(3, 0), 100))
References
[1]
Z. Kabluchko and D. Zaporozhets, Absorption probabilities for Gaussian polytopes and regular spherical simplices, Advances in Applied Probability 52 (2020), 588–616. (arXiv) (doi)
Links
Similar tables
External_solid_angles_of_regular_simplices —   The external angles of the same simplex face profiles measure normal cones instead of tangent cones.
Values_of_the_error_function —   The error function expresses the complex Gaussian distribution function in the angle integral.
Volume_of_Platonic_solids —   The regular tetrahedron is the three-dimensional simplex; this table gives volumes instead of angles.
Data properties
Entries are of type: real number
How well the digits are known: proven
Table is complete: no (it holds all 77 pairs $2\le d\le12$, $0\le k<d$, covering every proper-face type in each included simplex dimension; dimension and face dimension are discrete geometric indices, and the noncanonical cutoff 12 keeps certified quadrature, cancellation checks and independent high-precision verification practical while retaining complete face profiles)
How they were obtained:

The 11 facet values and the triangle vertex value are exact rationals. The other 65 entries use validated Arb quadrature in 500-bit ball arithmetic, with 167 guard bits beyond the requested 100 significant digits. The entire symmetrized imaginary-normal integrand is integrated over $[0,T]$, with an integer $T$ satisfying $aT^2/2\ge3(100+30)$, where $a=(k+1)/(d+1)$. The omitted tail is bounded by $\sqrt{2/\pi}\,e^{-aT^2/2}/(aT)$, and its outward upper endpoint is added to the radius. The attached numerical-method.md proves the bound. All resulting enclosures are positive and finite; the largest measured relative radius is less than $4.80\times10^{-123}$.

more

Every entry was recomputed at 140 requested digits. A separate algorithm computes the external angles using a real Gaussian integral in 630-bit ball arithmetic and recovers the internal angles through the conic Gauss–Bonnet recurrence (5). Its enclosures have relative radius below $10^{-140}$ and overlap all 77 results. Four nonzero controls for the external quadrature and the internal controls $1/6$ and $1/2$ are run first.

Exact projected Gram matrices verify the covariance for all 77 pairs. The checks also verify all 11 facet, 11 codimension-two and 10 codimension-three formulas, the equilateral triangle and regular tetrahedron identities, and Gram's alternating angle sum in each of the 11 dimensions. The conic recurrence is evaluated on all 77 values. The attached checks.py repeats the identities on every interval read back from the database; the attached generator verifies every entry. The mpmath program provides numerical approximations, while the attached generator supplies certified enclosures.



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