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Tridecagon
Tridecagon
Tridecagon
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Regular tridecagon
A regular tridecagon
TypeRegular polygon
Edges and vertices13
Schläfli symbol{13}
Coxeter–Dynkin diagrams
Symmetry groupDihedral (D13), order 2×13
Internal angle (degrees)≈152.308°
PropertiesConvex, cyclic, equilateral, isogonal, isotoxal
Dual polygonSelf

In geometry, a tridecagon or triskaidecagon or 13-gon is a thirteen-sided polygon.

Regular tridecagon

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A regular tridecagon is represented by Schläfli symbol {13}.

The measure of each internal angle of a regular tridecagon is approximately 152.308 degrees, and the area with side length a is given by

Construction

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As 13 is a Pierpont prime but not a Fermat prime, the regular tridecagon cannot be constructed using a compass and straightedge. However, it is constructible using neusis, or angle trisection.

The following is an animation from a neusis construction of a regular tridecagon with radius of circumcircle according to Andrew M. Gleason,[1] based on the angle trisection by means of the Tomahawk (light blue).

A neusis construction of a regular tridecagon (triskaidecagon) with radius of circumcircle as an animation (1 min 44 s), angle trisection by means of the Tomahawk (light blue). This construction is derived from the following equation:

Symmetry

[edit]
Symmetries of a regular tridecagon. Vertices are colored by their symmetry positions. Blue mirrors are drawn through vertices and edge. Gyration orders are given in the center.

The regular tridecagon has Dih13 symmetry, order 26. Since 13 is a prime number there is one subgroup with dihedral symmetry: Dih1, and 2 cyclic group symmetries: Z13, and Z1.

These 4 symmetries can be seen in 4 distinct symmetries on the tridecagon. John Conway labels these by a letter and group order.[2] Full symmetry of the regular form is r26 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars), and i when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as g for their central gyration orders.

Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g13 subgroup has no degrees of freedom but can be seen as directed edges.

Numismatic use

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The regular tridecagon is used as the shape of the Czech 20 korun coin.[3]

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A tridecagram is a 13-sided star polygon. There are 5 regular forms given by Schläfli symbols: {13/2}, {13/3}, {13/4}, {13/5}, and {13/6}. Since 13 is prime, none of the tridecagrams are compound figures.

Although 13-sided stars appear in the Topkapı Scroll, they are not of these regular forms.[4]

Petrie polygons

[edit]

The regular tridecagon is the Petrie polygon of the 12-simplex:

A12

12-simplex

References

[edit]
[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Tridecagon

View on Grokipedia
from Grokipedia
A tridecagon, also known as a triskaidecagon or 13-gon, is a consisting of thirteen sides and thirteen vertices. In its regular form, a tridecagon features thirteen equal-length sides and thirteen congruent interior angles, each measuring exactly 1980∘13\frac{1980^\circ}{13} or approximately 152.3077 degrees. The sum of its interior angles totals 1980 degrees, derived from the general formula for the interior angle sum of an n-sided , (n−2)×180∘(n-2) \times 180^\circ. Each exterior angle measures 360∘13\frac{360^\circ}{13} or approximately 27.6923 degrees. Unlike many regular polygons, a regular tridecagon cannot be constructed using only a and , as 13 is a not among the Fermat primes required for such constructibility. It exhibits the symmetry of the D13D_{13}, with 26 elements comprising 13 rotational symmetries and 13 reflectional symmetries. Irregular tridecagons, by contrast, may have unequal sides and angles while still enclosing a thirteen-sided boundary.

Definition and Basic Properties

Definition

A tridecagon is a polygon consisting of exactly 13 sides and 13 vertices, forming a closed plane figure. The name "tridecagon" is derived from Greek roots, with "trideka" meaning thirteen and "gonia" meaning angle, reflecting its 13-sided structure. Tridecagons may be classified as simple or complex; simple tridecagons do not self-intersect and can be either convex, where all interior angles are less than 180 degrees, or concave, with at least one interior angle greater than 180 degrees, whereas complex tridecagons feature self-intersections. In general, tridecagons are not required to have equal sides or angles, though a regular tridecagon possesses both equilateral and equiangular properties.

Fundamental Properties

A tridecagon is a with exactly 13 sides and 13 vertices, where the number of sides equals the number of vertices by definition. The sum of the interior angles of any simple tridecagon is given by the general formula for an nn-gon, (n−2)×180∘(n-2) \times 180^\circ, which for n=13n=13 yields 11×180∘=1980∘11 \times 180^\circ = 1980^\circ. In radians, this sum is (n−2)π=11π(n-2)\pi = 11\pi. This result holds for any simple regardless of regularity or convexity, derived from triangulating the polygon into n−2n-2 triangles, each contributing 180∘180^\circ or π\pi radians to the total. For simple tridecagons, each interior measures greater than 0∘0^\circ and less than 360∘360^\circ. In convex tridecagons, all interior angles are less than 180∘180^\circ, while concave variants have at least one interior angle between 180∘180^\circ and 360∘360^\circ. These bounds ensure the remains simple without self-intersections. The perimeter of a tridecagon is the sum of its 13 side lengths, with no assumptions of equality among the sides. In the special case of a regular tridecagon, all sides are equal, as detailed in the geometric features section.

Regular Tridecagon

Geometric Features

A regular tridecagon is a 13-sided where all sides are of equal length and all interior angles are equal, making it equilateral and equiangular. This uniformity results in a highly symmetric figure that appears nearly circular due to the large number of sides, yet retains distinct polygonal characteristics. The for the regular tridecagon is {13}, denoting its regular polygonal nature with 13 sides. As a , it lies entirely on one side of each of its edges, ensuring no internal angles exceed 180 degrees. It is cyclic, meaning all vertices lie on a common , and tangential, as it possesses an incircle tangent to all sides, a property true for all regular polygons. For a regular tridecagon with unit side length s=1s = 1, the circumradius RR (distance from center to a vertex) is given by R=12sin⁡(π/13),R = \frac{1}{2 \sin(\pi/13)}, and the rr (distance from center to the of a side) by r=12tan⁡(π/13).r = \frac{1}{2 \tan(\pi/13)}. These relations highlight the proportional , where the circumradius exceeds the apothem by a factor related to the of approximately 27.692 degrees per sector. Visually, the regular tridecagon cannot the without gaps or overlaps, as its interior s do not divide 360 degrees evenly, preventing seamless adjacency of multiple copies around a point. This contrasts with triangular, square, and hexagonal tilings, underscoring the tridecagon's role in more complex or non-periodic arrangements.

Angle and Side Calculations

The interior of a regular tridecagon is calculated using the for the interior angle of a regular nn-gon: (n−2n)×180∘\left( \frac{n-2}{n} \right) \times 180^\circ, where n=13n = 13. Substituting the value gives 1113×180∘=[1980](/page/1980)∘13≈152.308∘\frac{11}{13} \times 180^\circ = \frac{[1980](/page/1980)^\circ}{13} \approx 152.308^\circ. The exterior angle, which is the angle through which the turns at each vertex, is 360∘13≈27.692∘\frac{360^\circ}{13} \approx 27.692^\circ. This value also represents the subtended by each side at the center of the tridecagon, or 2π13\frac{2\pi}{13} radians when considering the inscribed in a . For a regular tridecagon inscribed in a (circumradius r=1r = 1), the side length ss is the chord length corresponding to the , given by s=2sin⁡(π13)s = 2 \sin\left( \frac{\pi}{13} \right). The area AA of a regular tridecagon with side length ss is A=134s2cot⁡(π13)A = \frac{13}{4} s^2 \cot\left( \frac{\pi}{13} \right). Alternatively, in terms of the circumradius rr, the area is A=132r2sin⁡(2π13)A = \frac{13}{2} r^2 \sin\left( \frac{2\pi}{13} \right).

Construction Methods

Theoretical Limitations

The construction of a regular tridecagon using only a and is theoretically impossible, a result stemming from the fact that 13 is a whose associated 2π/132\pi/13 is not a constructible . According to the Gauss-Wantzel theorem, a regular nn-gon is constructible n=2k∏pin = 2^k \prod p_i, where the pip_i are distinct Fermat primes; since 13 is prime but not a Fermat prime (the known Fermat primes being 3, 5, 17, 257, and ), the tridecagon falls outside this criterion. This theorem, articulated by in 1801 for sufficient conditions and rigorously completed by Pierre Wantzel in 1837 for necessity, relies on field theory to show that constructible lengths correspond to field extensions of degree a power of 2 over . The core obstruction lies in the minimal polynomial of cos⁡(2π/13)\cos(2\pi/13), which is irreducible over and has degree 6. This degree arises because the real subfield of the 13th cyclotomic field has degree ϕ(13)/2=6\phi(13)/2 = 6, where ϕ\phi is , and 6 is not a power of 2, preventing the coordinates of the tridecagon's vertices from being obtained through quadratic extensions alone. The 13th cyclotomic polynomial is Φ13(x)=x12+x11+⋯+x+1\Phi_{13}(x) = x^{12} + x^{11} + \cdots + x + 1, an of degree 12 that generates the Q(ζ13)\mathbb{Q}(\zeta_{13}), where ζ13=e2πi/13\zeta_{13} = e^{2\pi i /13}. Solving for the roots requires extensions beyond those achievable with and operations. This impossibility was established in the , paralleling the case of the regular (n=7), which also cannot be constructed exactly for similar reasons involving a degree-3 minimal for cos⁡(2π/7)\cos(2\pi/7). However, while some non-constructible regular polygons admit constructible star variants under relaxed conditions, the prime nature of 13 ensures that star tridecagons, such as {13/3} or {13/5}, share the same field-theoretic barriers and are likewise impossible with classical tools.

Practical Approximations

One practical method for approximating a regular tridecagon involves placing its vertices on the unit using parametric coordinates. The vertices are located at points (cos⁡(2πk13),sin⁡(2πk13))\left( \cos\left(\frac{2\pi k}{13}\right), \sin\left(\frac{2\pi k}{13}\right) \right) for k=0k = 0 to 1212, which can be computed numerically and plotted for visualization or further . In modern drawing applications, a regular tridecagon can be approximated with high precision using tools like protractors for manual sketching or (CAD) software for digital rendering. For instance, software such as allows users to draw regular polygons with up to 1024 sides via the command, specifying 13 sides and the circumradius or inscribed , yielding near-exact results limited only by computational precision. Historical approximations of the tridecagon have employed alternative tools beyond the and , such as the marked ruler () and folding, which enable greater accuracy for non-constructible polygons. Using a marked ruler, the tridecagon can be constructed exactly through techniques, as detailed in methods based on solving the associated cubic equations for the vertex angles. Similarly, origami methods allow for the folding of a regular tridecagon by simultaneously creasing multiple layers to achieve the required 360°/13 ≈ 27.692° divisions, offering practical accuracy superior to iterative compass divisions alone. Attempts to approximate a tridecagon using only a and , such as by iteratively marking arcs to divide the circle into 13 parts, introduce cumulative angular errors due to the inability to exactly trisect angles or solve the minimal of degree 6 over . For example, a common method involves setting the compass to a fraction of the (e.g., approximately 13/(2×13 + 1) = 0.46 times the ) and stepping around the ; this yields central deviations of about 0.25° per step, resulting in a total misalignment of roughly 3.25° after 13 steps, sufficient for visual but not precise applications.

Symmetry

Dihedral Symmetry Group

The dihedral symmetry group of the regular tridecagon is the D13D_{13}, which consists of all isometries that map the tridecagon to itself and has order 26. This group is generated by a rr by the 360∘13\frac{360^\circ}{13} and a reflection ss, encompassing 13 rotations and 13 reflections. The rotations are given by rkr^k for k=0,1,…,12k = 0, 1, \dots, 12, forming the cyclic subgroup ⟨r⟩\langle r \rangle of order 13. Since 13 is prime, the rotation subgroup ⟨r⟩\langle r \rangle is cyclic of prime order, possessing only the trivial subgroups {1}\{1\} and itself, which underscores its simple algebraic structure. The full dihedral group D13D_{13} includes the reflections rksr^k s for k=0,1,…,12k = 0, 1, \dots, 12, each corresponding to a reflection across one of the 13 axes of symmetry. For the odd-sided tridecagon, these axes each pass through one vertex and the midpoint of the opposite side, alternating in their geometric placement around the polygon. The abstract structure of D13D_{13} is captured by its group presentation ⟨r,s∣r13=1, s2=1, srs−1=r−1⟩\langle r, s \mid r^{13} = 1, \, s^2 = 1, \, s r s^{-1} = r^{-1} \rangle, where the relation srs−1=r−1s r s^{-1} = r^{-1} encodes how reflections conjugate rotations to their inverses. This presentation highlights the non-abelian nature of the group, as the rotations and reflections do not commute in general.

Visual Representations

Visual representations of the regular tridecagon often include diagrams that label its vertices and illustrate the 13 axes of , each passing through one vertex and the of the opposite side due to the odd number of sides. These diagrams highlight how the symmetries align the polygon onto itself under reflections, emphasizing the elements briefly referenced in discussions. Animations of the tridecagon typically depict continuous or discrete rotations by multiples of 360∘13≈27.692∘\frac{360^\circ}{13} \approx 27.692^\circ, demonstrating the 13 distinct rotational positions before returning to the original orientation. The dual polygon of a regular tridecagon is another regular tridecagon, rendering it self-dual among regular s with an odd number of sides. This self-duality is visually evident when the original polygon's vertices correspond to the dual's edge midpoints, producing an identical form rotated by half the . For edge-coloring visualizations, the tridecagon requires three colors to properly color its edges without adjacent edges sharing the same color, as it forms an odd-length C13C_{13} with chromatic index 3. Diagrams often show alternating colors with a third introduced to resolve the odd-cycle conflict, using patterns like two colors for 12 edges and the third for the closing edge.

Applications

Numismatic Designs

The tridecagon has been incorporated into modern numismatic designs primarily as the overall shape of coins, leveraging its distinctive 13-sided form for visual identification and symbolic purposes. The Czech Republic's 20 korun circulating coin, introduced in 1993 by the , is a prominent example of a regular tridecagon in everyday currency. Made of brass-plated steel with a of 26 mm and weight of 8.43 g, the coin features the crowned Czech lion on the obverse and Saint Wenceslas on horseback on the reverse, aiding in its differentiation from other denominations through the non-circular profile. Another notable instance is the Royal Canadian Mint's 2019 $50 fine silver honoring the monument on . This 3 oz. pure silver piece (99.99% fineness) adopts a shape to mirror the monument's 13-sided basin, which was updated in 2017 to include symbols for all 13 after Nunavut's addition. The coin's ultra-high relief design captures the in an antique finish, emphasizing national unity and the 1967 centennial of . With a limited mintage of 2,500, it represents a deliberate choice to evoke the monument's while highlighting Canada's federal structure. The design rationale for these tridecagon coins often ties into the number 13's broader symbolism, such as the approximate 13 lunar cycles in a solar year, which has historical roots in calendrical systems across cultures. In the Canadian example, the 13 sides specifically symbolize the completeness of Canada's provincial and territorial federation, promoting themes of unity and inclusivity rather than superstition. Although 13 is viewed as unlucky in some Western traditions—stemming from associations with events like the Last Supper or Norse mythology—these coin designs repurpose it positively for national identity. Minting tridecagon-shaped coins presents technical challenges due to the regular tridecagon's non-constructibility with classical compass and tools, as established by Pierre Wantzel in , since 13 is a prime not of the form 2^k + 1. As a result, modern mints employ numerical approximations—such as solving for vertex coordinates using or —to achieve near-regular sides and angles, ensuring uniformity in production via high-precision dies. This approach, briefly referencing practical approximation methods like iterative angle divisions, allows for accurate replication despite the geometric limitations.

Architectural and Artistic Uses

The tridecagon, owing to its non-constructibility with and , appears rarely in precise form within architecture but features in complex geometric designs through approximations or related star patterns. In , 13-point girih tiles—interlaced strapwork incorporating tridecagonal symmetry—were employed as early as the 11th century, as seen in the of the Barsian Mosque in , where sweeping 13-point patterns contribute to the intricate ornamental scheme. These designs highlight the tridecagon's role in evoking infinite repetition and cosmic order, a hallmark of Seljuk-era aesthetics. In artistic contexts beyond , tridecagonal motifs emerge in and modern digital creations, symbolizing cycles and transformation due to the 13's association with completeness and rarity. For instance, the tridecagram (a 13-pointed derived from the tridecagon) represents thresholds and fundamental patterns in esoteric art traditions. Digital artists leverage D_{13} dihedral symmetry—the tridecagon's rotational and reflectional group—to generate fractal-like kaleidoscopic visuals, producing non-repeating patterns that explore chaos and harmony in computational media. Such applications underscore the shape's conceptual appeal in contemporary design, where software enables exact renderings unattainable in traditional media.

Star Variants

A star tridecagon, also known as a tridecagram, is a non-convex isogonal polygon consisting of 13 equal sides and angles, formed by every k-th vertex of 13 equally spaced points on a . These are denoted by the {13/k}, where k is an from 2 to 6, as k=1 yields the convex tridecagon and higher k values produce enantiomorphs (mirror images). The parameter k represents the step or density, dictating the —the number of full rotations the path makes around the center—and the pattern of vertex connections, resulting in self-intersecting edges that create a star-shaped figure. The five distinct star variants are {13/2}, {13/3}, {13/4}, {13/5}, and {13/6}, each with increasing from 2 to 6, which measures the number of enclosed regions or the winding multiplicity of the boundary relative to the center. For a regular {n/k}, the is simply k when k < n/2 and gcd(n, k)=1, as is the case here since 13 is prime. This arises from the topological winding of the edges, where each increment in k adds more intersections and interior layers without altering the overall 13-fold . Visually, these star tridecagons differ markedly from the simple convex outline of the regular tridecagon by featuring dense interlacing paths of edges that cross multiple times, forming a complex, petal-like or spiky enclosure rather than a smooth boundary. For instance, {13/2} produces a sparsely intersecting with two layers of winding, while {13/6} exhibits the most intricate overlaps, approaching a near-circular form filled with six internal densities.

Petrie and Compound Forms

The regular tridecagon appears as the in several regular higher-dimensional , most notably the 12-simplex. A is defined as a closed skew polygon on the surface of a such that every pair of consecutive edges lies on a distinct face, but no three consecutive edges do. For the 12-simplex, this results in a regular 13-sided skew polygon that cycles through all 13 vertices of the in a non-planar path, providing a key projection for visualizing its structure. This property extends to other polytopes where the tridecagon serves as a in their orthogonal projections, highlighting the role of odd-sided polygons in higher-dimensional . The construction of such for simplices follows from the symmetry of the A_{12}, where the path alternates between facets in a manner that yields the 13-gon boundary. Regarding compound forms, the of sides in the tridecagon (13) precludes non-trivial regular compounds in the plane, as regular polygonal compounds require the parameter to allow decomposition into multiple isomorphic components sharing the same vertex set, which is impossible for prime-sided polygons without reducing to a single component. Unlike composite-sided polygons such as the enneagon {9/3}, which decomposes into three equilateral triangles, no such regular interweaving exists for tridecagons. In higher dimensions, tridecagons may participate in compounds as faces or sections of polytopes, but these are not direct compounds of multiple tridecagons.

References

  1. https://mathworld.wolfram.com/Tridecagon.html
  2. https://mathresearch.utsa.edu/wiki/index.php?title=Properties_of_Polygons_%28Sides%2C_Angles_and_Diagonals%29
  3. https://mathworld.wolfram.com/ConstructiblePolygon.html
  4. https://mathworld.wolfram.com/RegularPolygon.html
  5. https://testbook.com/maths/tridecagon
  6. https://www.2dcurves.com/line/linep.html
  7. https://en.wiktionary.org/wiki/tridecagon
  8. https://mathworld.wolfram.com/Polygon.html
  9. https://mathresearch.utsa.edu/wiki/index.php?title=Lines_%26_Angles
  10. https://connectedmath.msu.edu/families/glossary.aspx
  11. https://polytope.miraheze.org/wiki/Tridecagon
  12. http://math.geometryof.org/MNEG/etile.html
  13. https://faculty.etsu.edu/gardnerr/3040/Notes-Eves6/Eves6-14-2.pdf
  14. https://www.researchgate.net/publication/243025202_Why_was_Wantzel_overlooked_for_a_century_The_changing_importance_of_an_impossibility_result
  15. https://vdeangel.xula.edu/Publications/MinimalPoly.pdf
  16. https://www.isibang.ac.in/~sury/cyclotomy.pdf
  17. https://pressbooks.nvcc.edu/cad201/chapter/regular-polygons/
  18. https://doi.org/10.2307/2323624
  19. https://origamiusa.org/thefold/article/editorial-folding-tridecagon-odd-polygon
  20. https://geometrian.com/research/regular_polygons.html
  21. https://kconrad.math.uconn.edu/blurbs/grouptheory/dihedral.pdf
  22. https://cd1.edb.hkedcity.net/cd/maths/en/ref_res/material/MSS_e/Exemp09.pdf
  23. http://www.cs.toronto.edu/~stacho/cs137/cs137-lec4.pdf
  24. https://www.cnb.cz/en/banknotes-and-coins/coins/20-czk/
  25. https://www.mint.ca/en/shop/coins/2019/3-oz.-pure-silver-coin---centennial-flame---mintage-2500-2019
  26. https://www.canada.ca/en/public-services-procurement/services/infrastructure-buildings/parliamentary-precinct/discover/grounds.html
  27. https://www.britannica.com/topic/number-symbolism/13
  28. https://doi.org/10.1016/j.foar.2013.03.002
  29. https://paulohscwb.github.io/SacredGeometry/symbols/
  30. https://larryriddle.agnesscott.org/ifs/symmetric/symmetric.htm
  31. https://mathworld.wolfram.com/StarPolygon.html
  32. https://mathworld.wolfram.com/PetriePolygon.html
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