Hubbry Logo
List of polygonsList of polygonsMain
Open search
List of polygons
Community hub
List of polygons
logo
8 pages, 0 posts
0 subscribers
Be the first to start a discussion here.
Be the first to start a discussion here.
List of polygons
List of polygons
List of polygons
View on Wikipedia
from Wikipedia

A pentagon is a five-sided polygon. A regular pentagon has 5 equal edges and 5 equal angles.

In geometry, a polygon is traditionally a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed chain. These segments are called its edges or sides, and the points where two of the edges meet are the polygon's vertices (singular: vertex) or corners.

The word polygon comes from Late Latin polygōnum (a noun), from Greek πολύγωνον (polygōnon/polugōnon), noun use of neuter of πολύγωνος (polygōnos/polugōnos, the masculine adjective), meaning "many-angled". Individual polygons are named (and sometimes classified) according to the number of sides, combining a Greek-derived numerical prefix with the suffix -gon, e.g. pentagon, dodecagon. The triangle, quadrilateral and nonagon are exceptions, although the regular forms trigon, tetragon, and enneagon are sometimes encountered as well.

Greek numbers

[edit]

Polygons are primarily named by prefixes from Ancient Greek numbers.

English-Greek numbers[1][2]
English cardinal number English ordinal number Greek cardinal number Greek ordinal number
one first heis (fem. mia, neut. hen) protos
two second duo deuteros
three third treis tritos
four fourth tettares tetartos
five fifth pente pemptos
six sixth hex hektos
seven seventh hepta hebdomos
eight eighth okto ogdoös
nine ninth ennea enatos
ten tenth deka dekatos
eleven eleventh hendeka hendekatos
twelve twelfth dodeka dodekatos
thirteen thirteenth triskaideka dekatotritos
fourteen fourteenth tettareskaideka dekatotetartos
fifteen fifteenth pentekaideka dekatopemptos
sixteen sixteenth hekkaideka dekatohektos
seventeen seventeenth heptakaideka dekatohebdomos
eighteen eighteenth oktokaideka dekatoögdoös
nineteen nineteenth enneakaideka dekatoënatos
twenty twentieth eikosi eikostos
twenty-one twenty-first heiskaieikosi eikostoprotos
twenty-two twenty-second duokaieikosi eikostodeuteros
twenty-three twenty-third triskaieikosi eikostotritos
twenty-four twenty-fourth tetterakaieikosi eikostotetartos
twenty-five twenty-fifth pentekaieikosi eikostopemptos
twenty-six twenty-sixth hekkaieikosi eikostohektos
twenty-seven twenty-seventh heptakaieikosi eikostohebdomos
twenty-eight twenty-eighth oktokaieikosi eikostoögdoös
twenty-nine twenty-ninth enneakaieikosi eikostoënatos
thirty thirtieth triakonta triakostos
thirty-one thirty-first heiskaitriakonta triakostoprotos
forty fortieth tessarakonta tessarakostos
fifty fiftieth pentekonta pentekostos
sixty sixtieth hexekonta hexekostos
seventy seventieth hebdomekonta hebdomekostos
eighty eightieth ogdoëkonta ogdoëkostos
ninety ninetieth enenekonta enenekostos
hundred hundredth hekaton hekatostos
hundred and ten hundred and tenth dekakaihekaton hekatostodekatos
hundred and twenty hundred and twentieth ikosikaihekaton hekatostoikostos
two hundred two hundredth diakosioi diakosiostos
three hundred three hundredth triakosioi triakosiostos
four hundred four hundredth tetrakosioi tetrakosiostos
five hundred five hundredth pentakosioi pentakosiostos
six hundred six hundredth hexakosioi hexakosiostos
seven hundred seven hundredth heptakosioi heptakosiostos
eight hundred eight hundredth oktakosioi oktakosiostos
nine hundred nine hundredth enneakosioi enneakosiostos
thousand thousandth chilioi chiliostos
two thousand two thousandth dischilioi dischiliostos
three thousand three thousandth trischilioi trischiliostos
four thousand four thousandth tetrakischilioi tetrakischiliostos
five thousand five thousandth pentakischilioi pentakischiliostos
six thousand six thousandth hexakischilioi hexakischiliostos
seven thousand seven thousandth heptakischilioi heptakischiliostos
eight thousand eight thousandth oktakischilioi oktakischiliostos
nine thousand nine thousandth enneakischilioi enneakischiliostos
ten thousand ten thousandth myrioi myriastos
twenty thousand twenty thousandth dismyrioi dismyriastos
thirty thousand thirty thousandth trismyrioi trismyriastos
forty thousand forty thousandth tetrakismyrioi tetrakismyriastos
fifty thousand fifty thousandth pentakismyrioi pentakismyriastos
sixty thousand sixty thousandth hexakismyrioi hexakismyriastos
seventy thousand seventy thousandth heptakismyrioi heptakismyriastos
eighty thousand eighty thousandth oktakismyrioi oktakismyriastos
ninety thousand ninety thousandth enneakismyrioi enneakismyriastos
hundred thousand hundred thousandth dekakismyrioi dekakismyriastos
two hundred thousand two hundred thousandth ikosakismyrioi ikosakismyriastos
three hundred thousand three hundred thousandth triakontakismyrioi triakontakismyriastos
million millionth hekatontakismyrioi hekatontakismyriastos
two million two millionth diakosakismyrioi diakosakismyriastos
three million three millionth triakosakismyrioi triakosakismyriastos
ten million ten millionth chiliakismyrioi chiliakismyriastos
hundred million hundred millionth myriakismyrioi myriakismyriastos

Systematic polygon names

[edit]

To construct the name of a polygon with more than 20 and fewer than 100 edges, combine the prefixes as follows. The "kai" connector is not included by some authors.

Tens and Ones final suffix
-kai- 1 -hena- -gon
20 icosi- (icosa- when used alone) 2 -di-
30 triaconta- 3 -tri-
40 tetraconta- 4 -tetra-
50 pentaconta- 5 -penta-
60 hexaconta- 6 -hexa-
70 heptaconta- 7 -hepta-
80 octaconta- 8 -octa-
90 enneaconta- 9 -ennea-

Extending the system up to 999 is expressed with these prefixes.[3]

Polygon names
Ones Tens Twenties Thirties+ Hundreds
10 deca- 20 icosa- 30 triaconta-
1 hena- 11 hendeca- 21 icosi-hena- 31 triaconta-hena- 100 hecta-
2 di- 12 dodeca- 22 icosi-di- 32 triaconta-di- 200 dihecta-
3 tri- 13 triskaideca- 23 icosi-tri- 33 triaconta-tri- 300 trihecta-
4 tetra- 14 tetrakaideca- 24 icosi-tetra- 40 tetraconta- 400 tetrahecta-
5 penta- 15 pentakaideca- 25 icosi-penta- 50 pentaconta- 500 pentahecta-
6 hexa- 16 hexakaideca- 26 icosi-hexa- 60 hexaconta- 600 hexahecta-
7 hepta- 17 heptakaideca- 27 icosi-hepta- 70 heptaconta- 700 heptahecta-
8 octa- 18 octakaideca- 28 icosi-octa- 80 octaconta- 800 octahecta-
9 ennea- 19 enneakaideca- 29 icosi-ennea- 90 enneaconta- 900 enneahecta-

List of n-gons by Greek numerical prefixes

[edit]
List of n-gon names[4][5]
Sides Names
1 henagon monogon
2 digon bigon
3 trigon triangle
4 tetragon quadrilateral
5 pentagon
6 hexagon
7 heptagon septagon
8 octagon
9 enneagon nonagon
10 decagon
11 hendecagon undecagon
12 dodecagon
13 tridecagon triskaidecagon
14 tetradecagon tetrakaidecagon
15 pentadecagon pentakaidecagon
16 hexadecagon hexakaidecagon
17 heptadecagon heptakaidecagon septendecagon
18 octadecagon octakaidecagon
19 enneadecagon enneakaidecagon
20 icosagon
21 icosikaihenagon icosihenagon
22 icosikaidigon icosidigon icosadigon
23 icosikaitrigon icositrigon icosatrigon
24 icosikaitetragon icositetragon icosatetragon
25 icosikaipentagon icosipentagon icosapentagon
26 icosikaihexagon icosihexagon icosahexagon
27 icosikaiheptagon icosiheptagon icosaheptagon
28 icosikaioctagon icosioctagon icosaoctagon
29 icosikaienneagon icosienneagon icosaenneagon
30 triacontagon
31 triacontakaihenagon triacontahenagon tricontahenagon
32 triacontakaidigon triacontadigon tricontadigon
33 triacontakaitrigon triacontatrigon tricontatrigon
34 triacontakaitetragon triacontatetragon tricontatetragon
35 triacontakaipentagon triacontapentagon tricontapentagon
36 triacontakaihexagon triacontahexagon tricontahexagon
37 triacontakaiheptagon triacontaheptagon tricontaheptagon
38 triacontakaioctagon triacontaoctagon tricontaoctagon
39 triacontakaienneagon triacontaenneagon tricontaenneagon
40 tetracontagon tessaracontagon
41 tetracontakaihenagon tetracontahenagon tessaracontahenagon
42 tetracontakaidigon tetracontadigon tessaracontadigon
43 tetracontakaitrigon tetracontatrigon tessaracontatrigon
44 tetracontakaitetragon tetracontatetragon tessaracontatetragon
45 tetracontakaipentagon tetracontapentagon tessaracontapentagon
46 tetracontakaihexagon tetracontahexagon tessaracontahexagon
47 tetracontakaiheptagon tetracontaheptagon tessaracontaheptagon
48 tetracontakaioctagon tetracontaoctagon tessaracontaoctagon
49 tetracontakaienneagon tetracontaenneagon tessaracontaenneagon
50 pentacontagon pentecontagon
51 pentacontakaihenagon pentacontahenagon pentecontahenagon
52 pentacontakaidigon pentacontadigon pentecontadigon
53 pentacontakaitrigon pentacontatrigon pentecontatrigon
54 pentacontakaitetragon pentacontatetragon pentecontatetragon
55 pentacontakaipentagon pentacontapentagon pentecontapentagon
56 pentacontakaihexagon pentacontahexagon pentecontahexagon
57 pentacontakaiheptagon pentacontaheptagon pentecontaheptagon
58 pentacontakaioctagon pentacontaoctagon pentecontaoctagon
59 pentacontakaienneagon pentacontaenneagon pentecontaenneagon
60 hexacontagon hexecontagon
61 hexacontakaihenagon hexacontahenagon hexecontahenagon
62 hexacontakaidigon hexacontadigon hexecontadigon
63 hexacontakaitrigon hexacontatrigon hexecontatrigon
64 hexacontakaitetragon hexacontatetragon hexecontatetragon
65 hexacontakaipentagon hexacontapentagon hexecontapentagon
66 hexacontakaihexagon hexacontahexagon hexecontahexagon
67 hexacontakaiheptagon hexacontaheptagon hexecontaheptagon
68 hexacontakaioctagon hexacontaoctagon hexecontaoctagon
69 hexacontakaienneagon hexacontaenneagon hexecontaenneagon
70 heptacontagon hebdomecontagon
71 heptacontakaihenagon heptacontahenagon hebdomecontahenagon
72 heptacontakaidigon heptacontadigon hebdomecontadigon
73 heptacontakaitrigon heptacontatrigon hebdomecontatrigon
74 heptacontakaitetragon heptacontatetragon hebdomecontatetragon
75 heptacontakaipentagon heptacontapentagon hebdomecontapentagon
76 heptacontakaihexagon heptacontahexagon hebdomecontahexagon
77 heptacontakaiheptagon heptacontaheptagon hebdomecontaheptagon
78 heptacontakaioctagon heptacontaoctagon hebdomecontaoctagon
79 heptacontakaienneagon heptacontaenneagon hebdomecontaenneagon
80 octacontagon ogdoecontagon
81 octacontakaihenagon octacontahenagon ogdoecontahenagon
82 octacontakaidigon octacontadigon ogdoecontadigon
83 octacontakaitrigon octacontatrigon ogdoecontatrigon
84 octacontakaitetragon octacontatetragon ogdoecontatetragon
85 octacontakaipentagon octacontapentagon ogdoecontapentagon
86 octacontakaihexagon octacontahexagon ogdoecontahexagon
87 octacontakaiheptagon octacontaheptagon ogdoecontaheptagon
88 octacontakaioctagon octacontaoctagon ogdoecontaoctagon
89 octacontakaienneagon octacontaenneagon ogdoecontaenneagon
90 enneacontagon enenecontagon
91 enneacontakaihenagon enneacontahenagon enenecontahenagon
92 enneacontakaidigon enneacontadigon enenecontadigon
93 enneacontakaitrigon enneacontatrigon enenecontatrigon
94 enneacontakaitetragon enneacontatetragon enenecontatetragon
95 enneacontakaipentagon enneacontapentagon enenecontapentagon
96 enneacontakaihexagon enneacontahexagon enenecontahexagon
97 enneacontakaiheptagon enneacontaheptagon enenecontaheptagon
98 enneacontakaioctagon enneacontaoctagon enenecontaoctagon
99 enneacontakaienneagon enneacontaenneagon enenecontaenneagon
100 hectogon hecatontagon hecatogon
120 hecatonicosagon dodecacontagon
200 dihectagon diacosigon
300 trihectagon triacosigon
400 tetrahectagon tetracosigon
500 pentahectagon pentacosigon
600 hexahectagon hexacosigon
700 heptahectagon heptacosigon
800 octahectagon octacosigon
900 enneahectagon enneacosigon
1000 chiliagon
2000 dischiliagon dichiliagon
3000 trischiliagon trichiliagon
4000 tetrakischiliagon tetrachiliagon
5000 pentakischiliagon pentachiliagon
6000 hexakischiliagon hexachiliagon
7000 heptakischiliagon heptachiliagon
8000 octakischiliagon octachiliagon
9000 enneakischiliagon enneachilliagon
10000 myriagon
20000 dismyriagon dimyriagon
30000 trismyriagon trimyriagon
40000 tetrakismyriagon tetramyriagon
50000 pentakismyriagon pentamyriagon
60000 hexakismyriagon hexamyriagon
70000 heptakismyriagon heptamyriagon
80000 octakismyriagon octamyriagon
90000 enneakismyriagon enneamyriagon
100000 decakismyriagon decamyriagon
200000 icosakismyriagon icosamyriagon
300000 triacontakismyriagon tricontamyriagon
400000 tetracontakismyriagon tetracontamyriagon
500000 pentacontakismyriagon pentacontamyriagon
600000 hexacontakismyriagon hexacontamyriagon
700000 heptacontakismyriagon heptacontamyriagon
800000 octacontakismyriagon octacontamyriagon
900000 enneacontakismyriagon enneacontamyriagon
1000000 hecatontakismyriagon megagon
2000000 diacosakismyriagon dimegagon
3000000 triacosakismyriagon trimegagon
4000000 tetracosakismyriagon tetramegagon
5000000 pentacosakismyriagon pentamegagon
6000000 hexacosakismyriagon hexamegagon
7000000 heptacosakismyriagon heptamegagon
8000000 octacosakismyriagon octamegagon
9000000 enneacosakismyriagon enneamegagon
10000000 chiliakismyriagon decamegagon
20000000 dischiliakismyriagon icosamegagon
30000000 trischiliakismyriagon triacontamegagon
40000000 tetrakischiliakismyriagon tetracontamegagon
50000000 pentakischiliakismyriagon pentacontamegagon
60000000 hexakischiliakismyriagon hexacontamegagon
70000000 heptakischiliakismyriagon heptacontamegagon
80000000 octakischiliakismyriagon octacontamegagon
90000000 enneakischiliakismyriagon enneacontamegagon
100000000 myriakismyriagon hectamegagon
1000000000 gigagon[6]
∞ apeirogon

See also

[edit]

References

[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

List of polygons

View on Grokipedia
from Grokipedia
A list of polygons refers to the systematic classification and naming of two-dimensional closed geometric figures formed by connecting a finite number of straight line segments end-to-end, with each segment meeting its neighbors at vertices to create a bounded plane shape without self-intersections. These figures, essential in Euclidean geometry, are primarily distinguished by the number of sides (or edges), ranging from the equilateral triangle with three sides to polygons with dozens or more, where names become systematic for practicality. The nomenclature draws from Greek numerical prefixes combined with the suffix -gon (meaning "angle" or "corner"), reflecting the equal correspondence between sides and interior angles in simple polygons. Common polygons are named specifically for low side counts, while higher ones follow a pattern to accommodate mathematical and architectural applications. For instance: This naming extends up to 20 sides () and beyond, using multiplicative prefixes for very high numbers, such as myriagon for 10,000 sides. Beyond basic side-based lists, polygons are further categorized by properties like convexity (all interior angles less than 180°) or regularity (equal sides and angles), influencing their use in fields from tiling patterns to .

Basic Concepts in Polygon Nomenclature

Definition and Core Properties of Polygons

A is a closed plane figure consisting of a finite of line segments connected end-to-end, forming a closed chain, with vertices at the points where consecutive segments meet and no three successive vertices collinear. This applies to two-dimensional , where the figure lies entirely in a single plane and the boundary does not include curved lines. Polygons are classified based on several core properties. A simple polygon is one whose edges do not cross each other, meaning the only points shared by multiple edges are the vertices, resulting in a well-defined interior region. In contrast, a self-intersecting or complex polygon has edges that cross at points other than vertices, creating overlapping regions. Additionally, polygons can be convex or non-convex (concave): a convex polygon contains all line segments connecting any pair of its interior points, with every interior angle less than 180°, whereas a concave polygon has at least one interior angle greater than 180° and may include indentations. A regular polygon is both equilateral (all sides of equal length) and equiangular (all interior angles equal), exhibiting rotational symmetry around its center. The sum of the interior angles of a simple n-sided is given by the (n−2)×180∘(n-2) \times 180^\circ, derived by triangulating the into n−2n-2 triangles, each contributing 180∘180^\circ to the total, a result rooted in . For example, a (n=3) has an interior angle sum of 180∘180^\circ, illustrating the convex case where all angles are less than 180∘180^\circ, while a (n=4) sums to 360∘360^\circ and can be convex like a square or concave with one reflex angle. The term "polygon" originates from the Greek words polús (many) and gōnía (angle), meaning "many-angled," first appearing in English in the 1570s via Late Latin polygonum.

Classification by Side Count and Regularity

Polygons are primarily classified by the number of sides, denoted as nn-gons, where nn represents the count of edges forming the closed shape. For simple, non-degenerate polygons in Euclidean geometry, n≥3n \geq 3, ensuring the figure encloses a positive area without self-intersections. Degenerate cases exist for lower values, such as the digon with n=2n=2, which corresponds to a line segment and has zero area in the plane, though it appears in spherical geometry or as a limiting case. As nn increases, polygons approach a circle in shape, with arbitrary large nn allowing for highly approximate circular forms, but all maintain discrete sides. Within each nn-gon class, polygons are further categorized by regularity, which describes the uniformity of sides and angles. A has all sides of equal length (equilateral) and all interior angles equal (equiangular), resulting in full . An irregular polygon lacks this uniformity, featuring sides and angles of varying lengths and measures. Equilateral polygons have equal side lengths but varying angles, while equiangular polygons have equal angles but varying side lengths. Regularity significantly influences geometric properties, particularly for regular nn-gons, where enables precise formulas. The exterior angle at each vertex is 360∘n\frac{360^\circ}{n}, determining the turning angle when traversing the boundary. The , the distance from the center to a side, and the circumradius, the distance to a vertex, relate to the side ss via: s=2rsin⁡(πn)s = 2 r \sin\left(\frac{\pi}{n}\right) a=rcos⁡(πn)a = r \cos\left(\frac{\pi}{n}\right) where rr is the circumradius. These properties highlight how increasing nn refines the polygon's circular approximation, with the and radius converging. This classification by side count and regularity underpins polygon nomenclature, as regular nn-gons typically receive standardized names derived from numerical prefixes, facilitating concise identification. In contrast, irregular polygons often require supplementary descriptors to specify variations, such as "scalene" for a with unequal sides, emphasizing their non-uniform traits.

Etymology and Prefix Systems

Greek Numerical Prefixes

The naming of polygons with a small number of sides primarily relies on numerical prefixes derived from cardinal numbers, which are affixed to the suffix "-gon," originating from the Greek word γωνία (gōnía), meaning "" or "corner." These prefixes provide a systematic linguistic foundation for denoting the number of sides (n) in an n-gon, emphasizing the geometric focus on angles formed by those sides. The core Greek prefixes used in this nomenclature are as follows: mono- (1, though rarely applied to polygons due to conceptual challenges with a single-sided figure), di- or bi- (2), tri- (3), tetra- (4), penta- (5), hexa- (6), hepta- (7), octa- (8), ennea- (9; Latin nona-), deca- (10), hendeca- (11), and dodeca- (12). For numbers beyond 12, these prefixes can combine or extend, but the base forms establish the pattern for common polygons. Etymologically, these prefixes stem directly from Ancient Greek cardinal numerals, with adaptations for phonetic flow in compound words. For instance, "penta-" derives from πέντε (pénte, "five"), "hexa-" from ἕξ (héx, "six"), "hepta-" from ἑπτά (heptá, "seven"), "octa-" from ὀκτώ (oktṓ, "eight"), "ennea-" from ἐννέα (ennéa, "nine"), and "deca-" from δέκα (déka, "ten"); similar origins apply to the others, such as "tetra-" from τέσσαρες (téssares, "four"). In English adoption, these underwent minor phonetic shifts, such as vowel elision (e.g., "hexa-" to "hex-") or assimilation to Latin influences, while retaining their Greek roots for mathematical precision. When forming polygon names, the prefix precedes "-gon" without additional connectors, yielding terms like "pentagon" or "dodecagon." Exceptions include historical irregularities, such as "triangle" (a Latin-influenced variant of the Greek "trigon") and "octagon" (which favors the "-a-" spelling over the archaic "-o-" in "octogon"). The di-/bi- duality for two reflects interchangeable Greek (δύο, dúo) and Latin (bis) influences, though Greek forms predominate in pure geometric contexts. This system was standardized in mathematics, notably in Euclid's Elements (circa 300 BCE), where polygons are described by their angular counts using these prefixes, as in constructions of pentagons and hexagons in Book IV. The nomenclature gained widespread use during the , as European scholars translated and expanded upon classical Greek texts, integrating these terms into emerging vernacular geometries.

Systematic and Constructed Names

Systematic names for polygons extend the Greek prefix system to higher numbers of sides by combining numerical roots in a modular fashion, allowing for scalable nomenclature. For polygons with more than 20 sides, the name is typically formed by the prefix for the tens digit, optionally connected by "kai" (Greek for "and") to the prefix for the units digit, followed by the suffix "-gon." This approach uses established Greek roots such as "eikosi-" for 20, "triakonta-" for 30, and "tessarakonta-" for 40. For example, a 24-sided polygon is an icositetragon, combining "eikosi-" (20) and "tetra-" (4). Multiplicative prefixes like "di-," "tri-," and "tetra-" are incorporated to denote multiples, particularly for even-sided or repeated structures, enhancing the combinatorial flexibility of the system. For instance, names for numbers like 48 sides may employ "tessarakonta-" (40) with "octa-" (8), resulting in tessarakontaoctagon (or tetracontakaioctagon with "kai"), an additive combination of tens and units prefixes that simplifies breakdown for larger composites; a separate example of multiplicative usage is "dichiliagon" for 48 × something? Wait, no—for 2000 sides (2 × 1000), it is dichiliagon. For even larger polygons, specialized roots based on powers of ten are used, such as "pentekonta-" for 50, "hekaton-" for 100, "chilia-" for 1,000, and "myria-" for 10,000, appended directly to "-gon." A 1,000-sided polygon is thus a chiliagon, while a 10,000-sided one is a myriagon. These constructed names follow a rule-based structure analogous to systematic nomenclature in chemistry, prioritizing stems that reflect numerical composition over historical ad-hoc terms. As the number of sides increases significantly, the phonetic complexity and length of these names pose practical challenges, often resulting in abbreviations or the simple descriptor "n-gon" for clarity in mathematical discourse. This modular system contrasts with basic Greek prefixes for n ≤ 20 by enabling consistent extension to arbitrary sizes without inventing new roots.

Catalog of Named Polygons

Polygons with 3 to 12 Sides

The , also known as a trigon or 3-gon, is the simplest with three sides and three vertices. The sum of its interior angles is 180° for any , as derived from dividing the into basic triangular units. A has no diagonals, since the formula for the number of diagonals in an n-gon is n(n−3)2\frac{n(n-3)}{2}, yielding 0 for n=3. The regular , or , possesses the dihedral symmetry group D_3, consisting of 6 elements: 3 rotations and 3 reflections. Historically, the plays a central role in the , which relates the sides of a right-angled and was known to ancient Babylonians around 1900 BCE and formalized by around 500 BCE. In real-world applications, triangles provide structural stability in , such as in bridges and trusses, due to their rigidity. The , or 4-gon, features four sides and four vertices, with the sum of interior angles equaling 360°. It has 2 diagonals according to the standard formula. The regular quadrilateral, known as , has dihedral symmetry D_4 with 8 elements, including 4 rotations and 4 reflections. Quadrilaterals are fundamental in and , forming the basis for rectangles and squares used in building floors and windows. The , or 5-gon, has five sides and an interior angle sum of 540°. It contains 5 diagonals. The regular pentagon exhibits D_5 , with 10 elements. A prominent real-world example is building in Arlington, , headquarters of the U.S. Department of Defense, constructed in 1943 as a five-sided structure to efficiently house military operations. Regular pentagons appear in , such as in the arrangement of flower petals. The hexagon, or 6-gon, possesses six sides and an interior angle sum of 720°. It has 9 diagonals. The regular hexagon has D_6 symmetry, comprising 12 elements. Notably, regular hexagons tessellate the plane without gaps or overlaps, a property exploited in nature by honeybees to construct efficient honeycomb structures for storing honey and raising brood. This tessellation maximizes space usage, as proven in geometric packing theorems. The heptagon, or 7-gon, includes seven sides and an interior angle sum of 900°. It features 14 diagonals. The regular heptagon has D_7 symmetry with 14 elements. Unlike some lower-sided polygons, the regular heptagon cannot be constructed using only a compass and straightedge, as its vertices correspond to roots of an irreducible cubic equation over the rationals, a result from Galois theory. The octagon, or 8-gon, has eight sides and an interior angle sum of 1080°. It contains 20 diagonals. The regular octagon exhibits D_8 symmetry, with 16 elements. In traffic safety, octagonal shapes are uniquely used for stop signs in the United States and many other countries, as the eight-sided form was chosen in the early 20th century for its distinctiveness to command immediate attention and indicate the highest level of danger at intersections. The nonagon, or 9-gon, consists of nine sides with an interior angle sum of 1260°. It has 27 diagonals. The regular nonagon possesses D_9 symmetry, including 18 elements. While less common in everyday applications, nonagons appear in certain architectural designs, such as interlocking patterns in medieval Islamic structures like the Hagia Sophia, where they contribute to decorative geometric tilings. The , or 10-gon, features ten sides and an interior angle sum of 1440°. It includes 35 diagonals. The regular decagon has D_10 with 20 elements. Decagons are used in , as seen in medieval tilings and girih patterns that incorporate decagonal for ornamental motifs in mosques and madrasas. The undecagon, also called or 11-gon, has eleven sides and an interior angle sum of 1620°. It has 44 diagonals. The regular undecagon exhibits D_11 , comprising 22 elements. Due to its odd number of sides greater than 5, it shares the non-constructibility property with the under classical and methods. The , or 12-gon, includes twelve sides with an interior angle sum of 1800°. It contains 54 diagonals. The regular dodecagon has D_12 , with 24 elements, and is constructible with and . Historically, dodecagons relate to ancient divisions of the zodiac into twelve signs, influencing astronomical and astrological diagrams since Babylonian times around 700 BCE. In architecture, dodecagonal forms appear in and Roman designs, such as approximations in temple layouts and ornamental friezes, symbolizing completeness due to the number 12.

Polygons with 13 or More Sides

Polygons with 13 or more sides, known as n-gons where n ≥ 13, exhibit increasingly complex structures while adhering to systematic naming conventions derived from Greek numerical prefixes. These polygons are rarely encountered in practical applications due to their high vertex count, but they hold significant theoretical interest in and . As n grows, a regular n-gon inscribed in a of fixed approximates the circle more closely, with its perimeter approaching 2πr and area approaching πr², reflecting the limiting behavior where the side length becomes relative to the . The number of diagonals in an n-gon, which connect non-adjacent vertices, is given by the \frac{n(n-3)}{2}, leading to a quadratic increase in complexity; for example, a 13-gon has 65 diagonals, while a 20-gon has 170. Among these higher-sided polygons, constructibility with and varies based on whether n is a product of a power of 2 and distinct Fermat primes (3, 5, 17, 257, 65537). Specific examples include the (13 sides), (14 sides), (15 sides), (16 sides), (17 sides), (18 sides), enneadecagon (19 sides), and (20 sides).
nName (Alternative)Constructible?Number of Diagonals
13 (Triskaidecagon)No65
14 (Tetrakaidecagon)No77
15 (Pentakaidecagon)Yes90
16 (Hexakaidecagon)Yes104
17 (Heptakaidecagon)Yes119
18Octadecagon (Octakaidecagon)No135
19Enneadecagon (Enneakaidecagon)No152
20Yes170
The stands out for its historical and mathematical significance; in 1796, at age 19, proved it constructible by demonstrating that the 17th roots of unity could be obtained through successive quadratic extensions of the rationals, marking the first such proof for a beyond those known to the ancients. For even higher n, such as 21, systematic naming employs combined prefixes like icosikaihenagon (20 + 1). In , the vertices of regular high-n polygons correspond to the roots of unity, forming the basis for the used in and numerical methods. These properties underscore the transition from discrete polygonal forms to continuous circular limits as n increases.

Extensions and Variations

Star Polygons and Schläfli Symbols

Star polygons represent a class of non-convex polygons characterized by self-intersections, forming star-like shapes when regular. Unlike convex polygons, they connect vertices of a regular n-gon in a non-adjacent sequence, resulting in intersecting edges that create a dense interior region. These figures are a subset of polygons that exhibit self-intersections, extending the concept of simple polygons to more complex configurations. The provides a concise notation for regular , denoted as {n/k}, where n is the number of vertices and sides, and k is an integer greater than 1 that determines the connection step between vertices. For the symbol to represent a single, connected rather than a compound, k and n must be coprime (i.e., their is 1), and typically 1 < k < n/2 to avoid redundancy with the convex case {n/1} or its retrograde {n/(n-1)}. Introduced by Ludwig Schläfli in the 19th century as part of his work on regular polytopes, this notation captures the topological and geometric essence of these figures. Prominent examples include the pentagram, denoted {5/2}, which connects every second vertex of a regular pentagon, forming a five-pointed star with a density of 2. The heptagram {7/3} connects every third vertex of a heptagon, yielding a seven-pointed star, while {9/4} produces a nine-pointed figure by stepping four vertices at a time. Note that {6/2} simplifies to a compound of two triangles rather than a single star, illustrating how non-coprime values lead to decomposable structures. Key properties of regular star polygons include their density, which equals k and measures the number of enclosed regions or the winding multiplicity around the center. The winding number, closely related to density, quantifies how many times the polygon's boundary encircles the interior point, contributing to its topological complexity. Vertex angles in these polygons are equal across all points, forming acute tips that differ from the obtuse interiors of convex counterparts, with the specific measure depending on n and k but always maintaining rotational symmetry. Additionally, star polygons often appear in isogonal conjugate pairs, where {n/k} and its conjugate {n/(n-k)} share the same vertices but traverse in opposite directions, producing mirror-image or retrograded forms. Historically, star polygons like the pentagram have held significance in mysticism and occult traditions, symbolizing protection, the human form, or elemental forces in various esoteric practices from ancient times through the Renaissance.

Compound and Non-Simple Polygons

Compound polygons are regular geometric figures constructed as the union of two or more regular polygons or star polygons that share the same center and vertices, resulting in a multi-component structure distinct from single connected star polygons. These compounds arise when the integers n and k in the Schläfli symbol {n/k} are not coprime (i.e., gcd(n, k) > 1), resulting in multiple components rather than a single connected figure; for instance, the {6/2} consists of two equilateral triangles interlocked to form a six-pointed star. Similarly, the stellated , also known as the sedog and denoted {12/2}, is a compound of two regular hexagons rotated relative to each other, producing 12 edges and vertices in total. Another example is the of two pentagrams {10/4}, where two five-pointed stars {5/2} are combined, sharing vertices on a . Non-simple polygons extend beyond simple closed chains by allowing self-intersections, multiple boundary components, or interior holes, which complicate their topological properties compared to simple polygons. In two dimensions, examples include complex polygons with holes, such as annular regions bounded by outer and inner polygonal chains, used to model regions like lakes within islands in geographic data. Polyiamonds serve as non-simple analogs in the plane, formed by joining multiple equilateral triangles edge-to-edge, potentially creating shapes with irregular boundaries or perforations, analogous to polycubes in three dimensions which assemble cubes into more complex volumes. These structures are characterized by their connectivity, where non-adjacent edges may intersect, distinguishing them from simple polygons that do not cross themselves. Notation for compound and non-simple polygons often builds on extensions of three-dimensional concepts, such as the Kepler-Poinsot polyhedra, to two-dimensional forms; for example, the great dodecagram {12/5} represents a non-convex that can be viewed as part of such extensions, though it remains a single component. In practice, these polygons find applications in tiling patterns, where compounds like the enable aperiodic or symmetric tilings, and in artistic designs such as , which frequently incorporate interlocking stars and compounds for decorative motifs in and manuscripts. In computational geometry, non-simple polygons are essential for algorithms handling self-intersecting shapes, such as computing minimum perimeter enclosures or visibility graphs for in and geographic information systems.

References

  1. https://mathworld.wolfram.com/Polygon.html
  2. http://mathcentral.uregina.ca/qq/database/qq.09.96/rosa1.html
  3. https://byjus.com/maths/types-of-polygon/
  4. https://www.infoplease.com/math-science/mathematics/numbers-formulas/polygons-how-many-sides
  5. https://www.ducksters.com/kidsmath/polygons.php
  6. https://mathworld.wolfram.com/SimplePolygon.html
  7. https://mathworld.wolfram.com/ConvexPolygon.html
  8. https://mathworld.wolfram.com/RegularPolygon.html
  9. http://aleph0.clarku.edu/~djoyce/elements/bookI/propI32.html
  10. https://www.etymonline.com/word/polygon
  11. https://mathworld.wolfram.com/Digon.html
  12. https://mathresearch.utsa.edu/wiki/index.php?title=Definition_of_Polygons
  13. https://www.mathopenref.com/polygonirregular.html
  14. https://www.mathsisfun.com/geometry/regular-polygons.html
  15. https://www.cuemath.com/geometry/irregular-polygons/
  16. https://phrontistery.info/numbers.html
  17. https://www.georgehart.com/virtual-polyhedra/greek-prefixes.html
  18. http://hull-awe.org.uk/index.php/Prefixes_in_words_for_geometrical_figures
  19. http://aleph0.clarku.edu/~djoyce/elements/bookI/defI19.html
  20. https://explorable.com/greek-geometry
  21. https://mathworld.wolfram.com/Icositetragon.html
  22. https://ix23.com/mathematics/recreational-mathematics/names-for-polygons-groups-number-systems-tuples-and-greek-and-latin-prefixes/
  23. https://mathworld.wolfram.com/Myriagon.html
  24. https://www.ck12.org/flexi/geometry/construct-regular-polygons/how-many-diagonals-does-a-nonagon-a-decagon-and-an-n-gon-have/
  25. https://mathworld.wolfram.com/ConstructiblePolygon.html
  26. https://mathworld.wolfram.com/Heptadecagon.html
  27. https://www.ams.org/publicoutreach/feature-column/fcarc-geo-dft
  28. https://mathworld.wolfram.com/StarPolygon.html
  29. https://mathworld.wolfram.com/SchlaefliSymbol.html
  30. https://www.cs.cmu.edu/~kmcrane/Projects/WindingNumbersOnSurfaces/WindingNumbersOnSurfaces.pdf
  31. https://escholarship.org/content/qt8820h977/qt8820h977.pdf
  32. https://polytope.miraheze.org/wiki/Compound_of_two_hexagons
  33. https://www.scribd.com/document/748305664/PMP90
  34. https://mathworld.wolfram.com/Polyiamond.html
  35. https://math.stackexchange.com/questions/3497846/confusion-in-understanding-non-simple-polygons
  36. https://www.metmuseum.org/essays/geometric-patterns-in-islamic-art
  37. https://www.cs.cmu.edu/afs/cs/academic/class/15456-s14/Handouts/BKOS.pdf
Add your contribution
Related Hubs
User Avatar
No comments yet.