History of Kissing numbers $\tau_n$

 Display properties:   number-header: $\tau_n$-Numbers: []+Numbers:+- params:+    n: '1'+  number: '2'+  comment: $\tau_1=2$, attained by the integer lattice $\mathbb{Z}$.+- params:+    n: '2'+  number: '6'+  comment: $\tau_2=6$, attained by the hexagonal lattice $A_2$; a seventh circle cannot+    touch, since two of seven rays from the centre would meet at an angle smaller+    than $60^{\circ}$.+- params:+    n: '3'+  number: '12'+  comment: $\tau_3=12$, attained by the face-centred cubic lattice $A_3$ and by many+    other arrangements, among them the vertices of a regular icosahedron; whether+    a thirteenth sphere fits was disputed by Newton and Gregory in 1694, and the first+    complete proof that it does not is by Schütte and van der Waerden CITE{SvdW}.+- params:+    n: '4'+  number: '24'+  comment: $\tau_4=24$, attained by $D_4$, whose minimal vectors are the vertices+    of the 24-cell; proved by Musin CITE{Musin}, and the arrangement is unique up+    to isometry CITE{dLLdMK}.+- params:+    n: '5'+  number:+    args:+    - '[40, 44]'+    state:+      lower:+      - '18'+      upper:+      - 1c+  comment: $40$ is the kissing number of $D_5$ CITE{SPLAG}; the upper bound is the+    semidefinite programming bound of Mittelmann and Vallentin CITE{MV}.+- params:+    n: '6'+  number:+    args:+    - '[72, 77]'+    state:+      lower:+      - '28'+      upper:+      - 2d+  comment: $72$ is the kissing number of $E_6$ CITE{SPLAG}; the upper bound is by+    de Laat, Leijenhorst and de Muinck Keizer CITE{dLLdMK}.+- params:+    n: '7'+  number:+    args:+    - '[126, 134]'+    state:+      lower:+      - 3u+      upper:+      - '46'+  comment: $126$ is the kissing number of $E_7$ CITE{SPLAG}; the upper bound is by+    Mittelmann and Vallentin CITE{MV}.+- params:+    n: '8'+  number: '240'+  comment: $\tau_8=240$, attained by $E_8$; proved independently by Odlyzko and Sloane+    CITE{OS} and by Levenshtein CITE{Levenshtein}, and the arrangement is unique up+    to isometry CITE{BannaiSloane}. The theta series of $E_8$ is the Eisenstein series+    $E_4$, whose coefficient of $q$ is $240$.+  equals: HREF{Q-expansion_of_the_Eisenstein_series_E4#1}+- params:+    n: '9'+  number:+    args:+    - '[306, 363]'+    state:+      lower:+      - 9i+      upper:+      - bb+  comment: $306$ is attained by a nonlattice arrangement of Leech and Sloane CITE{LeechSloane};+    the largest kissing number of a lattice in dimension $9$ is $272$, that of $\Lambda_9$+    CITE{Watson}; the upper bound is by Machado and de Oliveira Filho CITE{MO}.+- params:+    n: '10'+  number:+    args:+    - '[510, 553]'+    state:+      lower:+      - fu+      upper:+      - h9+  comment: $510$ is attained by an arrangement of Ganzhinov CITE{Ganzhinov}; the laminated+    lattice $\Lambda_{10}$ has $336$ minimal vectors; the upper bound is by Machado+    and de Oliveira Filho CITE{MO}.+- params:+    n: '11'+  number:+    args:+    - '[604, 868]'+    state:+      lower:+      - is+      upper:+      - r4+  comment: $604$ is attained by an arrangement found in 2026 by AI agents on the EinsteinArena+    platform CITE{Bianchi}; $\Lambda_{11}$ has $438$ minimal vectors; the upper bound+    is by de Laat and Leijenhorst CITE{dLL}.+- params:+    n: '12'+  number:+    args:+    - '[841, 1355]'+    state:+      lower:+      - q9+      upper:+      - 1ab+  comment: $841$ is attained by an arrangement of Takhanov, Assylbekov and Yun CITE{TAY},+    one sphere more than the arrangements of size $840$ known before it; the Coxeter–Todd+    lattice $K_{12}$ has $756$ minimal vectors; the upper bound is by de Laat and+    Leijenhorst CITE{dLL}.+- params:+    n: '13'+  number:+    args:+    - '[1154, 2064]'+    state:+      lower:+      - '142'+      upper:+      - 20g+  comment: $1154$ is attained by an arrangement of Zinoviev and Ericson CITE{ZE};+    $\Lambda_{13}$ has $906$ minimal vectors; the upper bound is by de Laat and Leijenhorst+    CITE{dLL}.+- params:+    n: '14'+  number:+    args:+    - '[1932, 3174]'+    state:+      lower:+      - 1sc+      upper:+      - '336'+  comment: $1932$ is attained by an arrangement of Ganzhinov CITE{Ganzhinov}; $\Lambda_{14}$+    has $1422$ minimal vectors; the upper bound is by de Laat and Leijenhorst CITE{dLL}.+- params:+    n: '15'+  number:+    args:+    - '[2564, 4853]'+    state:+      lower:+      - 2g4+      upper:+      - 4nl+  comment: $2564$ is attained by a nonlattice arrangement of Leech and Sloane CITE{LeechSloane};+    $\Lambda_{15}$ has $2340$ minimal vectors; the upper bound is by de Laat and Leijenhorst+    CITE{dLL}.+- params:+    n: '16'+  number:+    args:+    - '[4320, 7320]'+    state:+      lower:+      - '470'+      upper:+      - 74o+  comment: $4320$ is the kissing number of the Barnes–Wall lattice $\Lambda_{16}$+    CITE{BarnesWall}; the upper bound is by de Laat and Leijenhorst CITE{dLL}.+- params:+    n: '17'+  number:+    args:+    - '[5730, 10978]'+    state:+      lower:+      - 5j2+      upper:+      - an2+  comment: $5730$ is attained by an arrangement of Cohn and Li CITE{CohnLi}, obtained+    by changing signs in the minimal vectors of a lattice; $\Lambda_{17}$ has $5346$+    minimal vectors; the upper bound is by de Laat and Leijenhorst CITE{dLL}.+- params:+    n: '18'+  number:+    args:+    - '[7654, 16406]'+    state:+      lower:+      - 7f6+      upper:+      - g0m+  comment: $7654$ is attained by an arrangement of Cohn and Li CITE{CohnLi}; $\Lambda_{18}$+    has $7398$ minimal vectors; the upper bound is by de Laat and Leijenhorst CITE{dLL}.+- params:+    n: '19'+  number:+    args:+    - '[11948, 24417]'+    state:+      lower:+      - blc+      upper:+      - nr1+  comment: $11948$ is attained by an arrangement of Ho CITE{Ho}, improving the $11692$+    of Cohn and Li CITE{CohnLi}; $\Lambda_{19}$ has $10668$ minimal vectors; the upper+    bound is by de Laat and Leijenhorst CITE{dLL}.+- params:+    n: '20'+  number:+    args:+    - '[19448, 36195]'+    state:+      lower:+      - ivo+      upper:+      - 13b3+  comment: $19448$ is attained by an arrangement of Cohn and Li CITE{CohnLi}; $\Lambda_{20}$+    has $17400$ minimal vectors; the upper bound is by de Laat and Leijenhorst CITE{dLL}.+- params:+    n: '21'+  number:+    args:+    - '[29768, 53524]'+    state:+      lower:+      - t28+      upper:+      - 1k8k+  comment: $29768$ is attained by an arrangement of Cohn and Li CITE{CohnLi}; $\Lambda_{21}$+    has $27720$ minimal vectors; the upper bound is by de Laat and Leijenhorst CITE{dLL}.+- params:+    n: '22'+  number:+    args:+    - '[49896, 80810]'+    state:+      lower:+      - 1gn8+      upper:+      - 2eta+  comment: $49896$ is the kissing number of the laminated lattice $\Lambda_{22}$ CITE{Leech};+    the upper bound is by de Laat and Leijenhorst CITE{dLL}.+- params:+    n: '23'+  number:+    args:+    - '[93150, 122351]'+    state:+      lower:+      - 2quu+      upper:+      - 3nff+  comment: $93150$ is the kissing number of the laminated lattice $\Lambda_{23}$ CITE{Leech};+    the upper bound is by de Laat and Leijenhorst CITE{dLL}.+- params:+    n: '24'+  number: '196560'+  comment: $\tau_{24}=196560$, attained by the Leech lattice $\Lambda_{24}$ CITE{Leech};+    proved independently by Odlyzko and Sloane CITE{OS} and by Levenshtein CITE{Levenshtein},+    and the arrangement is unique up to isometry CITE{BannaiSloane}.