قائمة مجسمات جونسون

|  J_{n} | اسم الشكل | صورة | الرؤوس | الحواف | الوجوه | مجموعة التناظر ورقمها الرتبة | مساحة السطح والحجم |
| --- | --- | --- | --- | --- | --- | --- | --- |
| 1 | هرم مربع متساوي الأضلاع |  | 5 | 8 | 5 |  C_{4v} من الرتبة 8 |  {\begin{aligned}A&=\left(1+{\sqrt {3}}\right)a^{2}\\&\approx 2.7321a^{2}\\V&={\frac {\sqrt {2}}{6}}a^{3}\\&\approx 0.2357a^{3}\end{aligned}} |
| 2 | هرم خماسي |  | 6 | 10 | 6 |  C_{5v} من الرتبة 10 |  {\begin{aligned}A&={\frac {a^{2}}{2}}{\sqrt {{\frac {5}{2}}\left(10+{\sqrt {5}}+{\sqrt {75+30{\sqrt {5}}}}\right)}}\\&\approx 3.8855a^{2}\\V&=\left({\frac {5+{\sqrt {5}}}{24}}\right)a^{3}\\&\approx 0.3015a^{3}\end{aligned}} |
| 3 | قبة مثلثية |  | 9 | 15 | 8 |  C_{3v} من الرتبة 6 |  {\begin{aligned}A&=\left(3+{\frac {5{\sqrt {3}}}{2}}\right)a^{2}\\&\approx 7.3301a^{2}\\V&=\left({\frac {5}{3{\sqrt {2}}}}\right)a^{3}\\&\approx 1.1785a^{3}\end{aligned}} |
| 4 | قبيبة مربعة  ‏ |  | 12 | 20 | 10 |  C_{4v} من الرتبة 8 |  {\begin{aligned}A&=\left(7+2{\sqrt {2}}+{\sqrt {3}}\right)a^{2}\\&\approx 11.5605a^{2}\\V&=\left(1+{\frac {2{\sqrt {2}}}{3}}\right)a^{3}\\&\approx 1.9428a^{3}\end{aligned}} |
| 5 | Pentagonal cupola |  | 15 | 25 | 12 |  C_{5v} من الرتبة 10 |  {\begin{aligned}A&=\left({\frac {1}{4}}\left(20+5{\sqrt {3}}+{\sqrt {5\left(145+62{\sqrt {5}}\right)}}\right)\right)a^{2}\\&\approx 16.5798a^{2}\\V&=\left({\frac {1}{6}}\left(5+4{\sqrt {5}}\right)\right)a^{3}\\&\approx 2.3241a^{3}\end{aligned}} |
| 6 | Pentagonal rotunda |  | 20 | 35 | 17 |  C_{5v} من الرتبة 10 |  {\begin{aligned}A&=\left({\frac {1}{2}}\left(5{\sqrt {3}}+{\sqrt {10\left(65+29{\sqrt {5}}\right)}}\right)\right)a^{2}\\&\approx 22.3472a^{2}\\V&=\left({\frac {1}{12}}\left(45+17{\sqrt {5}}\right)\right)a^{3}\\&\approx 6.9178a^{3}\end{aligned}} |
| 7 | Elongated triangular pyramid |  | 7 | 12 | 7 |  C_{3v} من الرتبة 6 |  {\begin{aligned}A&=\left(3+{\sqrt {3}}\right)a^{2}\\&\approx 4.7321a^{2}\\V&=\left({\frac {1}{12}}\left({\sqrt {2}}+3{\sqrt {3}}\right)\right)a^{3}\\&\approx 0.5509a^{3}\end{aligned}} |
| 8 | Elongated square pyramid |  | 9 | 16 | 9 |  C_{4v} من الرتبة 8 |  {\begin{aligned}A&=\left(5+{\sqrt {3}}\right)a^{2}\\&\approx 6.7321a^{2}\\V&=\left(1+{\frac {\sqrt {2}}{6}}\right)a^{3}\\&\approx 1.2357a^{3}\end{aligned}} |
| 9 | Elongated pentagonal pyramid |  | 11 | 20 | 11 |  C_{5v} من الرتبة 10 |  {\begin{aligned}A&={\frac {20+5{\sqrt {3}}+{\sqrt {25+10{\sqrt {5}}}}}{4}}a^{2}\\&\approx 8.8855a^{2}\\V&=\left({\frac {5+{\sqrt {5}}+6{\sqrt {25+10{\sqrt {5}}}}}{24}}\right)a^{3}\\&\approx 2.022a^{3}\end{aligned}} |
| 10 | Gyroelongated square pyramid |  | 9 | 20 | 13 |  C_{4v} من الرتبة 8 |  {\begin{aligned}A&=(1+3{\sqrt {3}})a^{2}\\&\approx 6.1962a^{2}\\V&={\frac {1}{6}}\left({\sqrt {2}}+2{\sqrt {4+3{\sqrt {2}}}}\right)a^{3}\\&\approx 1.1927a^{3}\end{aligned}} |
| 11 | Gyroelongated pentagonal pyramid |  | 11 | 25 | 16 |  C_{5v} من الرتبة 10 |  {\begin{aligned}A&={\frac {1}{4}}\left(15{\sqrt {3}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 8.2157a^{2}\\V&={\frac {1}{24}}\left(25+9{\sqrt {5}}\right)a^{3}\\&\approx 1.8802a^{3}\end{aligned}} |
| 12 | Triangular bipyramid |  | 5 | 9 | 6 |  D_{3h} من الرتبة 12 |  {\begin{aligned}A&={\frac {3{\sqrt {3}}}{2}}a^{2}\\&\approx 2.5981a^{2}\\V&={\frac {\sqrt {2}}{6}}a^{3}\\&\approx 0.2358a^{3}\end{aligned}} |
| 13 | هرم مزدوج خماسي  ‏ |  | 7 | 15 | 10 |  D_{5h} من الرتبة 20 |  {\begin{aligned}A&={\frac {5{\sqrt {3}}}{2}}a^{2}\\&\approx 4.3301a^{2}\\V&={\frac {1}{12}}\left(5+{\sqrt {5}}\right)a^{3}\\&\approx 0.603a^{3}\end{aligned}} |
| 14 | Elongated triangular bipyramid |  | 8 | 15 | 9 |  D_{3h} من الرتبة 12 |  {\begin{aligned}A&={\frac {3}{2}}\left(2+{\sqrt {3}}\right)a^{2}\\&\approx 5.5981a^{2}\\V&={\frac {1}{12}}\left(2{\sqrt {2}}+3{\sqrt {3}}\right)a^{3}\\&\approx 0.6687a^{3}\end{aligned}} |
| 15 | Elongated square bipyramid |  | 10 | 20 | 12 |  D_{4h} من الرتبة 16 |  {\begin{aligned}A&=2\left(2+{\sqrt {3}}\right)a^{2}\\&\approx 7.4641a^{2}\\V&={\frac {1}{3}}\left(3+{\sqrt {2}}\right)a^{3}\\&\approx 1.4714a^{3}\end{aligned}} |
| 16 | Elongated pentagonal bipyramid |  | 12 | 25 | 15 |  D_{5h} من الرتبة 20 |  {\begin{aligned}A&={\frac {5}{2}}\left(2+{\sqrt {3}}\right)a^{2}\\&\approx 9.3301a^{2}\\V&={\frac {1}{12}}\left(5+{\sqrt {5}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{3}\\&\approx 2.3235a^{3}\end{aligned}} |
| 17 | Gyroelongated square bipyramid |  | 10 | 24 | 16 |  D_{4d} من الرتبة 16 |  {\begin{aligned}A&=4{\sqrt {3}}a^{2}\\&\approx 6.9282a^{2}\\V&={\frac {1}{3}}\left({\sqrt {2}}+{\sqrt {4+3{\sqrt {2}}}}\right)a^{3}\\&\approx 1.4284a^{3}\end{aligned}} |
| 18 | Elongated triangular cupola |  | 15 | 27 | 14 |  C_{3v} من الرتبة 6 |  {\begin{aligned}A&={\frac {1}{2}}\left(18+5{\sqrt {3}}\right)a^{2}\\&\approx 13.3301a^{2}\\V&={\frac {1}{6}}\left(5{\sqrt {2}}+9{\sqrt {3}}\right)a^{3}\\&\approx 3.7766a^{3}\end{aligned}} |
| 19 | Elongated square cupola |  | 20 | 36 | 18 |  C_{4v} من الرتبة 8 |  {\begin{aligned}A&=(15+2{\sqrt {2}}+{\sqrt {3}})a^{2}\\&\approx 19.5605a^{2}\\V&=\left(3+{\frac {8{\sqrt {2}}}{3}}\right)a^{3}\\&\approx 6.7712a^{3}\end{aligned}} |
| 20 | Elongated pentagonal cupola |  | 25 | 45 | 22 |  C_{5v} من الرتبة 10 |  {\begin{aligned}A&={\frac {1}{4}}\left(60+5{\sqrt {3}}+10{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 26.5798a^{2}\\V&={\frac {1}{6}}\left(5+4{\sqrt {5}}+15{\sqrt {5+2{\sqrt {5}}}}\right)a^{3}\\&\approx 10.0183a^{3}\end{aligned}} |
| 21 | Elongated pentagonal rotunda |  | 30 | 55 | 27 |  C_{5v} من الرتبة 10 |  {\begin{aligned}A&={\frac {1}{2}}a^{2}\left(20+5{\sqrt {3}}+5{\sqrt {5+2{\sqrt {5}}}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)\\&\approx 32.3472a^{2}\\V&={\frac {1}{12}}a^{3}\left(45+17{\sqrt {5}}+30{\sqrt {5+2{\sqrt {5}}}}\right)\\&\approx 14.612a^{3}\end{aligned}} |
| 22 | Gyroelongated triangular cupola |  | 15 | 33 | 20 |  C_{3v} من الرتبة 6 |  {\begin{aligned}A&={\frac {1}{2}}\left(6+11{\sqrt {3}}\right)a^{2}\\&\approx 12.5263a^{2}\\V&={\frac {1}{3}}{\sqrt {{\frac {61}{2}}+18{\sqrt {3}}+30{\sqrt {1+{\sqrt {3}}}}}}a^{3}\\&\approx 3.5161a^{3}\end{aligned}} |
| 23 | Gyroelongated square cupola |  | 20 | 44 | 26 |  C_{4v} من الرتبة 8 |  {\begin{aligned}A&=(7+2{\sqrt {2}}+5{\sqrt {3}})a^{2}\\&\approx 18.4887a^{2}\\V&=\left(1+{\frac {2}{3}}{\sqrt {2}}+{\frac {2}{3}}{\sqrt {4+2{\sqrt {2}}+2{\sqrt {146+103{\sqrt {2}}}}}}\right)a^{3}\\&\approx 6.2108a^{3}\end{aligned}} |
| 24 | Gyroelongated pentagonal cupola |  | 25 | 55 | 32 |  C_{5v} من الرتبة 10 |  {\begin{aligned}A&={\frac {1}{4}}\left(20+25{\sqrt {3}}+10{\sqrt {5+2{\sqrt {5}}}}+{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 25.2400a^{2}\\V&=\left({\frac {5}{6}}+{\frac {2}{3}}{\sqrt {5}}+{\frac {5}{6}}{\sqrt {2{\sqrt {650+290{\sqrt {5}}}}-2{\sqrt {5}}-2}}\right)a^{3}\\&\approx 9.0733a^{3}\end{aligned}} |
| 25 | Gyroelongated pentagonal rotunda |  | 30 | 65 | 37 |  C_{5v} من الرتبة 10 |  {\begin{aligned}A&={\frac {1}{2}}\left(15{\sqrt {3}}+\left(5+3{\sqrt {5}}\right){\sqrt {5+2{\sqrt {5}}}}\right)a^{2}\\&\approx 31.0075a^{2}\\V&=\left({\frac {45}{12}}+{\frac {17}{12}}{\sqrt {5}}+{\frac {5}{6}}{\sqrt {2{\sqrt {650+290{\sqrt {5}}}}-2{\sqrt {5}}-2}}\right)a^{3}\\&\approx 13.6671a^{3}\end{aligned}} |
| 26 | Gyrobifastigium |  | 8 | 14 | 8 |  D_{2d} من الرتبة 8 |  {\begin{aligned}A&=\left(4+{\sqrt {3}}\right)a^{2}\\&\approx 5.7321a^{2}\\V&=\left({\frac {\sqrt {3}}{2}}\right)a^{3}\\&\approx 0.866a^{3}\end{aligned}} |
| 27 | Triangular orthobicupola |  | 12 | 24 | 14 |  D_{3h} من الرتبة 12 |  {\begin{aligned}A&=2\left(3+{\sqrt {3}}\right)a^{2}\\&\approx 9.4641a^{2}\\V&={\frac {5{\sqrt {2}}}{3}}a^{3}\\&\approx 2.357a^{3}\end{aligned}} |
| 28 | Square orthobicupola |  | 16 | 32 | 18 |  D_{4h} من الرتبة 16 |  {\begin{aligned}A&=2(5+{\sqrt {3}})a^{2}\\&\approx 13.4641a^{2}\\V&=\left(2+{\frac {4{\sqrt {2}}}{3}}\right)a^{3}\\&\approx 3.8856a^{3}\end{aligned}} |
| 29 | Square gyrobicupola |  | 16 | 32 | 18 |  D_{4d} من الرتبة 16 |  {\begin{aligned}A&=2(5+{\sqrt {3}})a^{2}\\&\approx 13.4641a^{2}\\V&=\left(2+{\frac {4{\sqrt {2}}}{3}}\right)a^{3}\\&\approx 3.8856a^{3}\end{aligned}} |
| 30 | Pentagonal orthobicupola |  | 20 | 40 | 22 |  D_{5h} من الرتبة 20 |  {\begin{aligned}A&=\left(10+{\sqrt {{\frac {5}{2}}\left(10+{\sqrt {5}}+{\sqrt {75+30{\sqrt {5}}}}\right)}}\right)a^{2}\\&\approx 17.7711a^{2}\\V&={\frac {1}{3}}\left(5+4{\sqrt {5}}\right)a^{3}\\&\approx 4.6481a^{3}\end{aligned}} |
| 31 | Pentagonal gyrobicupola |  | 20 | 40 | 22 |  D_{5d} من الرتبة 20 |  {\begin{aligned}A&=\left(10+{\sqrt {{\frac {5}{2}}\left(10+{\sqrt {5}}+{\sqrt {75+30{\sqrt {5}}}}\right)}}\right)a^{2}\\&\approx 17.7711a^{2}\\V&={\frac {1}{3}}\left(5+4{\sqrt {5}}\right)a^{3}\\&\approx 4.6481a^{3}\end{aligned}} |
| 32 | Pentagonal orthocupolarotunda |  | 25 | 50 | 27 |  C_{5v} من الرتبة 10 |  {\begin{aligned}A&=\left(5+{\frac {1}{4}}{\sqrt {1900+490{\sqrt {5}}+210{\sqrt {75+30{\sqrt {5}}}}}}\right)a^{2}\\&\approx 23.5385a^{2}\\V&={\frac {5}{12}}\left(11+5{\sqrt {5}}\right)a^{3}\\&\approx 9.2418a^{3}\end{aligned}} |
| 33 | Pentagonal gyrocupolarotunda |  | 25 | 50 | 27 |  C_{5v} من الرتبة 10 |  {\begin{aligned}A&=\left(5+{\frac {15}{4}}{\sqrt {3}}+{\frac {7}{4}}{\sqrt {25+10{\sqrt {5}}}}\right)a^{2}\\&\approx 23.5385a^{2}\\V&={\frac {5}{12}}\left(11+5{\sqrt {5}}\right)a^{3}\\&\approx 9.2418a^{3}\end{aligned}} |
| 34 | Pentagonal orthobirotunda |  | 30 | 60 | 32 |  D_{5h} من الرتبة 20 |  {\begin{aligned}A&=\left((5{\sqrt {3}}+3{\sqrt {5\left(5+2{\sqrt {5}}\right)}}\right)a^{2}\\&\approx 29.306a^{2}\\V&={\frac {1}{6}}(45+17{\sqrt {5}})a^{3}\\&\approx 13.8355a^{3}\end{aligned}} |
| 35 | Elongated triangular orthobicupola |  | 18 | 36 | 20 |  D_{3h} من الرتبة 12 |  {\begin{aligned}A&=2(6+{\sqrt {3}})a^{2}\\&\approx 15.4641a^{2}\\V&=\left({\frac {5{\sqrt {2}}}{3}}+{\frac {3{\sqrt {3}}}{2}}\right)a^{3}\\&\approx 4.9551a^{3}\end{aligned}} |
| 36 | Elongated triangular gyrobicupola |  | 18 | 36 | 20 |  D_{3d} من الرتبة 12 |  {\begin{aligned}A&=2(6+{\sqrt {3}})a^{2}\\&\approx 15.4641a^{2}\\V&=\left({\frac {5{\sqrt {2}}}{3}}+{\frac {3{\sqrt {3}}}{2}}\right)a^{3}\\&\approx 4.9551a^{3}\end{aligned}} |
| 37 | Elongated square gyrobicupola |  | 24 | 48 | 26 |  D_{4d} من الرتبة 16 |  {\begin{aligned}A&=2(9+{\sqrt {3}})a^{2}\\&\approx 21.4641a^{2}\\V&=\left(4+{\frac {10{\sqrt {2}}}{3}}\right)a^{3}\\&\approx 8.714a^{3}\end{aligned}} |
| 38 | Elongated pentagonal orthobicupola |  | 30 | 60 | 32 |  D_{5h} من الرتبة 20 |  {\begin{aligned}A&=\left(20+{\sqrt {{\frac {5}{2}}\left(10+{\sqrt {5}}+{\sqrt {75+30{\sqrt {5}}}}\right)}}\right)a^{2}\\&\approx 27.7711a^{2}\\V&={\frac {1}{6}}\left(10+8{\sqrt {5}}+15{\sqrt {5+2{\sqrt {5}}}}\right)a^{3}\\&\approx 12.3423a^{3}\end{aligned}} |
| 39 | Elongated pentagonal gyrobicupola |  | 30 | 60 | 32 |  D_{5d} من الرتبة 20 |  {\begin{aligned}A&=\left(20+{\sqrt {{\frac {5}{2}}\left(10+{\sqrt {5}}+{\sqrt {75+30{\sqrt {5}}}}\right)}}\right)a^{2}\\&\approx 27.7711a^{2}\\V&={\frac {1}{6}}\left(10+8{\sqrt {5}}+15{\sqrt {5+2{\sqrt {5}}}}\right)a^{3}\\&\approx 12.3423a^{3}\end{aligned}} |
| 40 | Elongated pentagonal orthocupolarotunda |  | 35 | 70 | 37 |  C_{5v} من الرتبة 10 |  {\begin{aligned}A&={\frac {1}{4}}\left(60+{\sqrt {10\left(190+49{\sqrt {5}}+21{\sqrt {75+30{\sqrt {5}}}}\right)}}\right)a^{2}\\&\approx 33.5385a^{2}\\V&={\frac {5}{12}}\left(11+5{\sqrt {5}}+6{\sqrt {5+2{\sqrt {5}}}}\right)a^{3}\\&\approx 16.936a^{3}\end{aligned}} |
| 41 | Elongated pentagonal gyrocupolarotunda |  | 35 | 70 | 37 |  C_{5v} من الرتبة 10 |  {\begin{aligned}A&={\frac {1}{4}}\left(60+{\sqrt {10\left(190+49{\sqrt {5}}+21{\sqrt {75+30{\sqrt {5}}}}\right)}}\right)a^{2}\\&\approx 33.5385a^{2}\\V&={\frac {5}{12}}\left(11+5{\sqrt {5}}+6{\sqrt {5+2{\sqrt {5}}}}\right)a^{3}\\&\approx 16.936a^{3}\end{aligned}} |
| 42 | Elongated pentagonal orthobirotunda |  | 40 | 80 | 42 |  D_{5h} من الرتبة 20 |  {\begin{aligned}A&=\left(10+{\sqrt {30\left(10+3{\sqrt {5}}+{\sqrt {75+30{\sqrt {5}}}}\right)}}\right)a^{2}\\&\approx 39.306a^{2}\\V&={\frac {1}{6}}\left(45+17{\sqrt {5}}+15{\sqrt {5+2{\sqrt {5}}}}\right)a^{3}\\&\approx 21.5297a^{3}\end{aligned}} |
| 43 | Elongated pentagonal gyrobirotunda |  | 40 | 80 | 42 |  D_{5d} من الرتبة 20 |  {\begin{aligned}A&=\left(10+{\sqrt {30\left(10+3{\sqrt {5}}+{\sqrt {75+30{\sqrt {5}}}}\right)}}\right)a^{2}\\&\approx 39.306a^{2}\\V&={\frac {1}{6}}\left(45+17{\sqrt {5}}+15{\sqrt {5+2{\sqrt {5}}}}\right)a^{3}\\&\approx 21.5297a^{3}\end{aligned}} |
| 44 | Gyroelongated triangular bicupola |  | 18 | 42 | 26 |  D_{3} من الرتبة 6 |  {\begin{aligned}A&=\left(6+5{\sqrt {3}}\right)a^{2}\\&\approx 14.6603a^{2}\\V&={\sqrt {2}}\left({\frac {5}{3}}+{\sqrt {1+{\sqrt {3}}}}\right)a^{3}\\&\approx 4.6946a^{3}\end{aligned}} |
| 45 | Gyroelongated square bicupola |  | 24 | 56 | 34 |  D_{4} من الرتبة 8 |  {\begin{aligned}A&=\left(10+6{\sqrt {3}}\right)a^{2}\\&\approx 20.3923a^{2}\\V&=\left(2+{\frac {4}{3}}{\sqrt {2}}+{\frac {2}{3}}{\sqrt {4+2{\sqrt {2}}+2{\sqrt {146+103{\sqrt {2}}}}}}\right)a^{3}\\&\approx 8.1536a^{3}\end{aligned}} |
| 46 | Gyroelongated pentagonal bicupola |  | 30 | 70 | 42 |  D_{5} من الرتبة 10 |  {\begin{aligned}A&={\frac {1}{2}}\left(20+15{\sqrt {3}}+{\sqrt {25+10{\sqrt {5}}}}\right)a^{2}\\&\approx 26.4313a^{2}\\V&=\left({\frac {5}{3}}+{\frac {4}{3}}{\sqrt {5}}+{\frac {5}{6}}{\sqrt {2{\sqrt {650+290{\sqrt {5}}}}-2{\sqrt {5}}-2}}\right)a^{3}\\&\approx 11.3974a^{3}\end{aligned}} |
| 47 | Gyroelongated pentagonal cupolarotunda |  | 35 | 80 | 47 |  C_{5} من الرتبة 5 |  {\begin{aligned}A&={\frac {1}{4}}\left(20+35{\sqrt {3}}+7{\sqrt {25+10{\sqrt {5}}}}\right)a^{2}\\&\approx 32.1988a^{2}\\V&=\left({\frac {55}{12}}+{\frac {25}{12}}{\sqrt {5}}+{\frac {5}{6}}{\sqrt {2{\sqrt {650+290{\sqrt {5}}}}-2{\sqrt {5}}-2}}\right)a^{3}\\&\approx 15.9911a^{3}\end{aligned}} |