凸正多邊形多面體是指所有面皆由正多邊形組成的凸多面體,包括了無窮多種的棱柱體和反棱柱、柏拉圖立體、阿基米德立體和詹森多面體。一般對於凸正多邊形多面體的研究通常不討論無窮集合的棱柱體和反棱柱,不包括棱柱體和反棱柱的話,凸正多邊形多面體共有110個[4],若嚴格不計棱柱體與反棱柱,則立方體(正四角柱)和正八面體(正三角反棱柱)不算,共108個[4];有些文獻會將除了柏拉圖立體、阿基米德立體和詹森多面體外的邊數最少之棱柱體和反棱柱也列入(正三角柱和正四角反棱柱)共有112個。[5]柏拉圖立體與阿基米德立體早在公元前就已被發現,而詹森多面體發現得則較晚,由諾曼·詹森(英语:Norman Johnson (mathematician))在1966年發現並命名,並由維克托·查加勒(英语:Victor Zalgaller)在1969年證明諾曼·詹森所列出的立體是完整的,沒其其他更多立體有此特性,至此,嚴格凸的正多邊形多面體研究已算完備。[4]
非凸正多邊形多面體有無限多種,其中星形均勻多面體經常被研究探討,所有星形均勻多面體皆是非凸正多邊形多面體,但星形均勻多面體通常具有自相交的面。無自相交面的非凸正多邊形多面體亦有無限多種,目前尚未有系統性的分類。邦妮·斯圖爾特(Bonnie Stewart)在其著作《環形體歷險記》(Adventures among the Toroids)中探討了關於環形多面體的非凸正多邊形多面體。[16][4]
^Freudenthal, H; van der Waerden, B. L., Over een bewering van Euclides ("On an Assertion of Euclid"), Simon Stevin, 1947, 25: 115–128 (Dutch) 引文格式1维护:未识别语文类型 (link)(其表明只有8個凸正三角面多面體)
^Introducing the Kasparian Solids. quantimegroup.com. [2019-09-27]. (原始内容于2018-08-31).
^ 4.04.14.24.34.4Robert R Tupelo-Schneck. Regular-faced Polyhedra.
^Martin Berman. Regular-faced convex polyhedra. Journal of the Franklin Institute. 1971-05, 291 (5): 329–336 [2023-02-04]. doi:10.1016/0016-0032(71)90071-8(英语).
^Ivanov, B. A. Polyhedra with boundary surfaces compounded from regular polygons. Ukrainskiĭ Geometricheskiĭ Sbornik. 1971, 10: 20–34. ISSN 0135-6992(Russian). 引文格式1维护:未识别语文类型 (link)
^Prjahin, Ju. A. Convex polyhedra with regular faces. Ukrainskiĭ Geometricheskiĭ Sbornik. 1973, (No. 14): 83–88 (Russian). 引文格式1维护:未识别语文类型 (link)
^Steve Waterman. Convex hulls having regular diamonds.
^Gurin, AM and Zalgaller, VA. On the history of the study of convex polyhedra with regular faces and faces composed of regular ones. Translations of the American Mathematical Society-Series 2. 2009, 228: 169.
^ 11.011.1Timofeenko, Aleksei Victorovich. Junction of noncomposite polygons. Algebra i Analiz (St. Petersburg Department of Steklov Institute of Mathematics, Russian~…). 2009, 21 (3): 165–209.
^Timofeenko, Aleksei Victorovich. Corrections to “Junction of noncomposite polyhedra”. St. Petersburg Mathematical Journal. 2012-08-01, 23 (4): 779–780 [2023-01-31]. ISSN 1061-0022. doi:10.1090/S1061-0022-2012-01217-3(英语).
^Robert R Tupelo-Schneck. Convex regular-faced polyhedra with conditional edges.
^Yu. A. Pryakhin. Convex polyhedra whose faces are equiangular or composed of such. Journal of Soviet Mathematics. 1978-09, 10 (3): 486–487 [2023-02-04]. ISSN 0090-4104. doi:10.1007/BF01476855(英语).
^A. V. Timofeenko. Convex polyhedra with parquet faces. Doklady Mathematics. 2009-10, 80 (2): 720–723 [2023-02-04]. ISSN 1064-5624. doi:10.1134/S1064562409050238(英语).
^Bonnie M. Stewart. Adventures Among the Toroids 2nd ed. 1980. 引文格式1维护:冗余文本 (link)