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External solid angles of regular simplices
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Numbers
$d$
$k$ 
$\gamma_{d,k}$
2
0:
1/3
2
1:
1/2
3
0:
1/4
3
1:
0.3040867239846963649145722203887845443416856752802998563560308509889992956612788765648940869121127435
3
2:
1/2
4
0:
1/5
4
1:
0.2061300859770445473718583305831768165125285129204497845340462764834989434919183148473411303681691152
4
2:
0.2902153116275831218854750966642421277920322156237683791001392166912535932797984765762747494259325509
4
3:
1/2
5
0:
1/6
5
1:
0.1497376529171844216740635258094946032168806119146163367246394288102858201479182306554272193406295401
5
2:
0.1853229674413746828282126449963631916880483234356525686502088250368803899196977148644121241388988264
5
3:
0.2820471084244874658059371430847574678928978867431218675492038277924214868971577141378126183852732281
5
4:
1/2
6
0:
1/7
6
1:
0.1141273223312201418987282635517751471879873415857912009215214445532163112165983852263328310712919916
6
2:
0.1264792494455821645337838032084399850477089564289652155092431282960191660109733863686126396189557794
6
3:
0.1730706626367311987089057146271362018393468301146828013238057416886322303457365712067189275779098421
6
4:
0.2766501895190568373456625773352752524028854553119323286284957406780020795298379750434151547723356400
6
5:
1/2
7
0:
1/8
7
1:
0.09011663730715084719120004249389112828170300555955024218834488047849670835540514659723182245486002058
7
2:
0.09065984454499760662077176636049464313919185201299209102275977901191393182793727448084472548255807120
7
3:
0.1130125447054570946116122351590475964922148627946087019802959966871766654271051526588515934067836756
7
4:
0.1649752842785852560184938660029128786043281829678984929427436110170031192947569625651227321585034600
7
5:
0.2728144741714949688584304152229449013054226401657250670892020976231073193843908478700551638580338814
7
6:
1/2
8
0:
1/9
8
1:
0.07311436938897610712415261197777888642595475564868139305693601496923015739306169878671112193868540265
8
2:
0.06748272571048178673164878600647085287495026502357776603537966665282535785242499582258213342445183236
8
3:
0.07741359070242407201418773018572532149829244012871708607773042223688794632486859630259743755377611896
8
4:
0.1042240479189976443771654529063758117367142067212861431690709784937966590817588592422151517483586802
8
5:
0.1592217112572424532876456228344173519581339602485876006338031464346609790765862718050827457870508220
8
6:
0.2699465438373841147862194322948571069900886047292858979441250125544730063330707075278012016723071727
8
7:
1/2
9
0:
1/10
9
1:
0.06060818229689291342003481705328840304139215304730057008406423110707318054957824681067897908043275061
9
2:
0.05175687942746509368765098792334045771121082287864452596941323595288377932535435870456173929201012817
9
3:
0.05508322023504029321356455535213541733644812999043033984792009999258524260858028246541457838858000576
9
4:
0.06893464599985201757875718892775139833457187852338453635143809453948644887980221049699999244005760056
9
5:
0.09803671550747122208053015907380317159259366269274250378365765096679933517939575330119635748874100527
9
6:
0.1549198157560761721793291484422856604851329070939288469161875188317095094996060612917018025084607591
9
7:
0.2677204728012300107452337959412318914943660025783258458145956319036707174219725619389220629466750613
9
8:
1/2
10
0:
1/11
10
1:
0.05112518583919696887618305062422648029502596364018768183897046432295898824060850618370430399292309596
10
2:
0.04067368655867911455484601760371211804894234258552143508006109575139307187806474657951733409824775650
10
3:
0.04042614310700983935820888848076090424611621631645965640677057078167813906301690131103207509604175291
10
4:
0.04732969720171699955975695248910053859287190921390721501616409348885573536437737039618534196951095179
10
5:
0.06305560333994063305524935962714567571792755631754359173613888335923003948751226357785298407676780979
10
6:
0.09344504706336240553869957007080231915984764817333902259913167262903854992501542559840697025381529504
10
7:
0.1515807092018450161178506939118478372415490038674887687218934478555060761329588429083830944200125920
10
8:
0.2659421402146299619792080802886955347936438258375825407436094989817705068375464450666639240399114108
10
9:
1/2
11
0:
1/12
11
1:
0.04375349703473753676516724350248325117396642226148496397162202435061406931667395158526109297269145985
11
2:
0.03261572515550603552171728039801290276148473594624094703896893694024388133710633047118116643900038156
11
3:
0.03044142567187841317088319505671241120184450071227361171021512734491262924655506224919462552988433914
11
4:
0.03352051086487690228840868280145368260909965421271095042640854250929614187322951597312524420467409931
11
5:
0.04205399744929569597622624902629482564634689673197019491444854604930099753232250180877445304228507388
11
6:
0.05874625806494572688120034443986303042439760861013281163751665018641385897986179517651442145377030572
11
7:
0.08990264332263614893892147903038614281864934509359653207362380911805758883710390847779907428674861090
11
8:
0.1489132103219449429688121204330433021904657387563738111154142484726557602563196675999958860598671162
11
9:
0.2644886349179281212920929250719772765647437241191401479527374924269062075165893683957354962511754186
11
10:
1/2
12
0:
1/13
12
1:
0.03790302067414486553496385943509543452229236482404614674684000620331648618277594119285064131426563863
12
2:
0.02660319333380196625090467992529204706687339511553090980107694467446051170001826325449344356228228210
12
3:
0.02342531255811252816149808439641437154573754361771947313420093837970291110437133431265474687826902414
12
4:
0.02437083126354363278151641130880580152252200461587231558020694032410782257653431973999306306848777581
12
5:
0.02891970918285891572567233953717560203395731174051197853018749025568616207951981758082167813086971668
12
6:
0.03824650837790109124490659818207151748153772373514315624098328468233331945014609015873926681737114842
12
7:
0.05545542630351534548421920772288562831070835628817671564757044303596717853782836634719252835018387394
12
8:
0.08708696779608032511083557142922820298513725555475540832854951221194366832212216539831163826982579878
12
9:
0.1467329523768921819381393876079659148471155861787102219291062386403593112748840525936032443767631278
12
10:
0.2632783106866643301177528568334100962735266015517980976256235251145186937496163525453812954404922359
12
11:
1/2
Definition
For the regular simplex $S_d=\operatorname{conv}(e_0,\ldots,e_d)\subset\mathbb R^{d+1}$, regarded in its own affine hull, $\gamma_{d,k}$ is the external solid angle at any one $k$-dimensional face [1]. It is the standard Gaussian measure of the normal cone in its own $(d-k)$-dimensional linear span.
Parameters
$d$
—   simplex dimension ($d\ge2$)
$k$
—   face dimension ($0\le k<d$)
Formulas
(1)
$\displaystyle \gamma_{d,k}=g_{d-k}\!\left(\frac1{k+1}\right)=\frac1{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-x^2/2}\Phi\!\left(\frac{x}{\sqrt{k+1}}\right)^{d-k}\,\mathrm dx.$ Here $g_m(r)=\Pr(X_1<0,\ldots,X_m<0)$ for a centered Gaussian vector with covariance $I_m+r\mathbf1\mathbf1^T$, and $\Phi(t)=(1+\operatorname{erf}(t/\sqrt2))/2$, where $\operatorname{erf}$ is the error function. The correlation between distinct coordinates is $r/(1+r)=1/(k+2)$. This is Proposition 1.2(a) and equation (1.19) of [1].
(2)
$\displaystyle \gamma_{d,0}=\frac1{d+1},\qquad \gamma_{d,d-1}=\frac12.$
(3)
$\displaystyle \gamma_{d,d-2}=\frac{\arccos(-1/d)}{2\pi}=\frac14+\frac1{2\pi}\arcsin\!\left(\frac1d\right).$ The codimension-two normal cone is a planar wedge. The inverse trigonometric functions take their usual real principal values.
(4)
$\displaystyle \gamma_{d,d-3}=\frac18+\frac3{4\pi}\arcsin\!\left(\frac1{d-1}\right),\qquad d\geq3.$ This is the spherical-triangle case, equivalently Proposition 1.4(f) of [1].
Comments
(5)
The full sphere has angular measure $1$. Write $V=\{v\in\mathbb R^{d+1}:\sum_i v_i=0\}$ for the direction space of $S_d$. If $x$ is in the relative interior of the face $F$, its normal cone is $N_F=\{u\in V:\langle u,y-x\rangle\leq0\text{ for every }y\in S_d\}$. Thus the angle is taken in $\operatorname{span}N_F$, not in $V$ when the cone has smaller dimension.
(6)
A $k$-face has $k+1$ vertices. Every face of that dimension has the same angle, and multiplication by $\binom{d+1}{k+1}$ gives the sum over all such faces. Uniform scaling of the simplex leaves these angles unchanged. In [2], the simplex $\Delta_n$ has $n$ vertices and the face index counts vertices; its indices are therefore $n=d+1$ and $k_{\mathrm{source}}=k+1$.
(7)
The normal angles at the vertices sum to $1$, since the vertex normal cones partition the direction space up to their boundaries. The normal cone at a facet is a ray, whose angle in its one-dimensional span is $1/2$.
Programs
(P1)
Python
# Numerical approximations; the attached generator certifies the digits.
import mpmath as mp
from fractions import Fraction

def external_angle(d, k, digits=100):
    if not (d >= 2 and 0 <= k < d):
        raise ValueError("require d >= 2 and 0 <= k < d")
    if k == 0:
        return Fraction(1, d+1)
    if k == d-1:
        return Fraction(1, 2)
    with mp.workdps(digits+30):
        q = mp.mpf(k+1)
        m = d-k
        def f(t):
            return mp.exp(-q*t*t)*((mp.erfc(-t)/2)**m
                                  + (mp.erfc(t)/2)**m)
        return mp.sqrt(q/mp.pi)*mp.quad(f, [0,1,2,4,8,mp.inf])

# For all entries in this table:
# [(d, k, external_angle(d, k)) for d in range(2, 13) for k in range(d)]
References
[1]
Z. Kabluchko and D. Zaporozhets, Absorption probabilities for Gaussian polytopes and regular spherical simplices, Advances in Applied Probability 52 (2020), 588–616. (doi)
Links
Similar tables
Values_of_the_error_function —   The error function expresses the standard Gaussian distribution function in the one-dimensional angle integral.
Volume_of_Platonic_solids —   The regular tetrahedron is the three-dimensional member of the simplex family; 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 $2\leq d\leq12$, $0\leq k<d$, covering every face type in each included simplex dimension; dimension and face dimension are discrete geometric indices, and the noncanonical cutoff $12$ makes full certified quadrature and independent high-precision checks practical while retaining complete face profiles)
How they were obtained:

The 22 vertex and facet entries are exact rationals. The other 55 entries are enclosed using Arb quadrature of the entire Gaussian integrand in 413-bit ball arithmetic, with 80 guard bits beyond the 100 significant digits requested. Integration is over $[-25,25]$. Both omitted tails together are bounded by $2e^{-25^2/2}/(25\sqrt{2\pi})$ because the power of $\Phi$ lies between $0$ and $1$ on the real axis. The outward upper bound for this tail is added to the radius. Every returned ball is finite and has relative radius less than $10^{-110}$; the largest measured relative radius is $2.94\times10^{-114}$.

more

An independent implementation uses mpmath tanh-sinh quadrature of the transformed integral at 125 and 155 decimal digits for all 77 entries. The two precisions agree within $10^{-119}$, and the largest absolute difference from an Arb midpoint is less than $7.14\times10^{-124}$. These computations check agreement; the Arb enclosures and explicit tail bound certify the stored precision. Both implementations first pass the nonzero normalization controls $1/2$, $1/3$ and $1/8$.

Exact normal Gram-matrix calculations check the covariance for every entry. The planar wedge and spherical-excess formulas check all 11 codimension-two and 10 codimension-three cases. The 11 vertex and 11 facet identities and 30 applicable reflection identities are also checked. The attached checks.py reproduces these tests and can repeat them on every interval read from the database. The displayed mpmath program was executed on all 77 pairs and agrees with the certified values.



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