The kissing number in dimension 5: is it 40, or can it exceed 40 up to 44?
Difficulty 64: the SDP bound has stalled at 44 for 17 years and no extension of Musin's dimension-4 technique is known, but the question is finite and concrete with active configuration work; it sits below union-closed sets (66) and well below high-dimensional sphere packing (76), above cap sets (58). Application value 4.5: spherical codes matter for channel coding, but τ₅ itself would not change practice, far below sphere packing (26). Leverage 19: little depends on the value, though a four-point SDP method would transfer to other dimensions.
The problem
How many equal balls can touch one central ball in five dimensions? Between 40 and 44; nobody knows the exact number.
- 1Balls must touch one central ball
- 2Dimension 5: 40 balls arranged
- 3Maths limit: at most 44 can fit
- 4Nobody knows if 41 to 44 are possible
Statement. Let be the maximum number of non-overlapping unit spheres in that can touch a central unit sphere. Equivalently, it is the maximum size of a set of unit vectors in with pairwise inner products at most , i.e. angular separation at least (a ). The root lattice gives and semidefinite programming gives . determine : either prove or construct a spherical code of size at least with minimal angle .
Path to understanding
foundations → problemThe research entries this problem rests on, in the order to read them. Each links to its explainer.
- Start with
- Step 2
- Harmonic analysisTheory
- Fast matrix multiplicationAlgorithm
- Step 3
- Linear programmingAlgorithm
- Step 4
- Lattices and sphere packingTheory · host
- Then
- The kissing number in dimension 5: is it 40, or can it exceed 40 up to 44?The problem
State of the art
The kissing number is known exactly only in dimensions 1, 2, 3, 4, 8 and 24; τ₄ = 24 is due to Musin (Annals 2008). In dimension 5 the bounds 40 ≤ τ₅ ≤ 44 have not moved since the Mittelmann–Vallentin high-accuracy SDP computation (arXiv:0902.1105; Experiment. Math. 2010), which brought the Bachoc–Vallentin three-point bound from 45 to 44 (the SDP value is 44.998…). Four non-isometric 40-point kissing configurations are now known: D₅ (Korkine–Zolotareff, 1873), L₅ (Leech, 1967), Q₅ (Szöllősi, arXiv:2301.08272, 2023) and R₅, found by Cohn and Rajagopal (arXiv:2412.00937, revised March 2026), who say τ₅ appears to be 40. Recent SDP advances (Leijenhorst–de Laat, arXiv:2202.12077) and AI-driven searches improved bounds in other dimensions, not in dimension 5; Cohn's table still lists 40–44 as of October 2026. The coexistence of several inequivalent 40-point configurations, unlike the unique optima in dimensions 8 and 24, is a structural puzzle.
Realistic applications
Kissing configurations are optimal spherical codes, relevant to code design for continuous channels and quantisation; a determination of tau_5 would have no direct deployment impact but would test SDP and polynomial-optimisation hierarchies.
What counts as solved
A proof (with rigorous exact certificate, e.g. an exact SDP or exclusion argument) that tau_5 = 40, or an explicit set of at least 41 unit vectors in R^5 with pairwise inner products at most 1/2, which can be verified exactly.
How it gets verified
Bound / recordA numeric bound or record checked from a certificate or recomputation.
Quantity: the kissing number tau_5, currently between 40 and 44. An improved lower bound is an explicit set of at least 41 unit vectors in R^5 with pairwise inner products at most 1/2, verified in exact arithmetic; an upper bound of 40 needs an exact semidefinite-programming or exclusion certificate. Record the vectors or certificate and the exact checker.
Claims under verification
+ add a claimNo claimed results on the Verification board for this problem yet.
Where the work is
Either a new bounding technique (theory) or a heavy exact SDP and rounding computation; configuration searches are also computational.
Bottlenecks
ranked by votes- #1TheorySDP relaxations leave a gap of 4
Three-point SDP bounds give 44 (the SDP value is 44.998…); a rigorous 40 would need higher-order (Lasserre-type) constraints or a new argument that accommodates several distinct optimal-looking configurations, unlike the unique optima in dimensions 8 and 24.
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Progress
- Mar 4, 2026Fourth 40-point configuration, revised papersource
Cohn and Rajagopal analyse four non-isometric 40-point configurations; none beats 40.
- Jan 19, 2023Third 40-point configuration (Szöllősi)source
A new kissing arrangement Q₅, not isometric to D₅ or L₅.
- Feb 6, 2009SDP bound improved to 44source
Mittelmann and Vallentin reduce the upper bound on τ₅ from 45 to 44 (arXiv date).