4158:
818:
447:
40:
4312:
246:
273:: any line drawn through the polygon (and not tangent to an edge or corner) meets its boundary exactly twice. As a consequence, all its interior angles are less than 180°. Equivalently, any line segment with endpoints on the boundary passes through only interior points between its endpoints. This condition is true for polygons in any geometry, not just Euclidean.
2635:
4228:
2661:
Considering the enclosed regions as point sets, we can find the area of the enclosed point set. This corresponds to the area of the plane covered by the polygon or to the area of one or more simple polygons having the same outline as the self-intersecting one. In the case of the cross-quadrilateral,
4425:
The imaging system calls up the structure of polygons needed for the scene to be created from the database. This is transferred to active memory and finally, to the display system (screen, TV monitors etc.) so that the scene can be viewed. During this process, the imaging system renders polygons in
2425:
150:
is one which does not intersect itself. More precisely, the only allowed intersections among the line segments that make up the polygon are the shared endpoints of consecutive segments in the polygonal chain. A simple polygon is the boundary of a region of the plane that is called a
1400:
2657:
of the region. For example, the central convex pentagon in the center of a pentagram has density 2. The two triangular regions of a cross-quadrilateral (like a figure 8) have opposite-signed densities, and adding their areas together can give a total area of zero for the whole
1189:
5021:
2467:
5690:
2104:
3015:
2843:
4426:
correct perspective ready for transmission of the processed data to the display system. Although polygons are two-dimensional, through the system computer they are placed in a visual scene in the correct three-dimensional orientation.
3967:
As with René Descartes's example of the chiliagon, the million-sided polygon has been used as an illustration of a well-defined concept that cannot be visualised. The megagon is also used as an illustration of the convergence of
780:, so the sum of the exterior angles must be 360°. This argument can be generalized to concave simple polygons, if external angles that turn in the opposite direction are subtracted from the total turned. Tracing around an
403:
The property of regularity may be defined in other ways: a polygon is regular if and only if it is both isogonal and isotoxal, or equivalently it is both cyclic and equilateral. A non-convex regular polygon is called a
951:
2303:
1203:
1013:
3231:
3150:
5687:
1547:-axis. If the vertices are ordered counterclockwise (that is, according to positive orientation), the signed area is positive; otherwise, it is negative. In either case, the area formula is correct in
1625:
580:
3428:
Exceptions exist for side counts that are easily expressed in verbal form (e.g. 20 and 30), or are used by non-mathematicians. Some special polygons also have their own names; for example the
2269:
709:
758:
615:
294:: the whole interior is visible from at least one point, without crossing any edge. The polygon must be simple, and may be convex or concave. All convex polygons are star-shaped.
4924:
276:
Non-convex: a line may be found which meets its boundary more than twice. Equivalently, there exists a line segment between two boundary points that passes outside the polygon.
4381:
2630:{\displaystyle A={\frac {ns^{2}}{4}}\cot {\frac {\pi }{n}}={\frac {ns^{2}}{4}}\cot {\frac {\alpha }{n-2}}=n\cdot \sin {\frac {\alpha }{n-2}}\cdot \cos {\frac {\alpha }{n-2}}.}
4483:
2167:
652:
3991:
To construct the name of a polygon with more than 20 and fewer than 100 edges, combine the prefixes as follows. The "kai" term applies to 13-gons and higher and was used by
1531:
1482:
1620:
43:
Some polygons of different kinds: open (excluding its boundary), boundary only (excluding interior), closed (including both boundary and interior), and self-intersecting.
3342:
is a three-dimensional solid bounded by flat polygonal faces, analogous to a polygon in two dimensions. The corresponding shapes in four or higher dimensions are called
2849:
2677:
2459:
1436:
2116:
gives a simple formula for the polygon's area based on the numbers of interior and boundary grid points: the former number plus one-half the latter number, minus 1.
4709:
3038:
3386:, the masculine adjective), meaning "many-angled". Individual polygons are named (and sometimes classified) according to the number of sides, combining a
5839:, Provides an interactive Java investigation that extends the interior angle sum formula for simple closed polygons to include crossed (complex) polygons
2653:
Using the formulas for simple polygons, we allow that particular regions within the polygon may have their area multiplied by a factor which we call the
2420:{\displaystyle A=R^{2}\cdot {\frac {n}{2}}\cdot \sin {\frac {2\pi }{n}}=R^{2}\cdot n\cdot \sin {\frac {\pi }{n}}\cdot \cos {\frac {\pi }{n}}}
2671:
Using the same convention for vertex coordinates as in the previous section, the coordinates of the centroid of a solid simple polygon are
828:
3521:
The simplest polygon which can cross itself; the simplest polygon which can be concave; the simplest polygon which can be non-cyclic. Can
5678:
Reprint of original 1921 publication with corrected errata. Heath uses the
Latinized spelling "Aristophonus" for the vase painter's name.
1395:{\displaystyle 16A^{2}=\sum _{i=0}^{n-1}\sum _{j=0}^{n-1}{\begin{vmatrix}Q_{i,j}&Q_{i,j+1}\\Q_{i+1,j}&Q_{i+1,j+1}\end{vmatrix}},}
2179:
The lengths of the sides of a polygon do not in general determine its area. However, if the polygon is simple and cyclic then the sides
5274:
Arthur
Baragar (2002) Constructions Using a Compass and Twice-Notched Straightedge, The American Mathematical Monthly, 109:2, 151–164,
3425:
Beyond decagons (10-sided) and dodecagons (12-sided), mathematicians generally use numerical notation, for example 17-gon and 257-gon.
1184:{\displaystyle A={\frac {1}{2}}\sum _{i=0}^{n-1}(x_{i}y_{i+1}-x_{i+1}y_{i})\quad {\text{where }}x_{n}=x_{0}{\text{ and }}y_{n}=y_{0},}
3253:, a polygon having only two sides and two corners, which is impossible in a flat plane. Spherical polygons play an important role in
3283:
is an infinite sequence of sides and angles, which is not closed but has no ends because it extends indefinitely in both directions.
4157:
6970:
3297:
is an area-connected or multiply-connected planar polygon with one external boundary and one or more interior boundaries (holes).
3156:
3075:
776:-gon, the angle "turned" at a corner is the exterior or external angle. Tracing all the way around the polygon makes one full
5671:
5427:
5239:
5079:
4742:
4683:
530:
triangles, each of which has an angle sum of π radians or 180 degrees. The measure of any interior angle of a convex regular
5830:
5761:
Schirra, Stefan (2008). "How
Reliable Are Practical Point-in-Polygon Strategies?". In Halperin, Dan; Mehlhorn, Kurt (eds.).
3331:
representing the various elements (sides, vertices, etc.) and their connectivity. A real geometric polygon is said to be a
784:-gon in general, the sum of the exterior angles (the total amount one rotates at the vertices) can be any integer multiple
6405:
5818:
5259:
Benjamin, Elliot; Snyder, C. Mathematical
Proceedings of the Cambridge Philosophical Society 156.3 (May 2014): 409–424.;
4716:
537:
5864:
5806:
4656:
297:
183:
2224:
1539:
of the plane. Commonly, the positive orientation is defined by the (counterclockwise) rotation that maps the positive
6992:
5644:
5621:
5583:
5560:
5537:
5514:
5476:
5453:
5444:
4351:
3335:
of the associated abstract polygon. Depending on the mapping, all the generalizations described here can be realized.
5697:, Castellani Halls, Capitoline Museum, accessed 2013-11-11. Two pentagrams are visible near the center of the image,
5505:
5490:
5747:
476:
Any polygon has as many corners as it has sides. Each corner has several angles. The two most important ones are:
4844:
4673:
4380:, as well as other attributes of the polygon, such as color, shading and texture), connectivity information, and
3936:
4611:
4587:
4499:
4333:
4698:
B.Sz. Nagy, L. Rédey: Eine
Verallgemeinerung der Inhaltsformel von Heron. Publ. Math. Debrecen 1, 42–50 (1949)
3472:
Not generally recognised as a polygon, although some disciplines such as graph theory sometimes use the term.
665:
5763:
Algorithms - ESA 2008: 16th Annual
European Symposium, Karlsruhe, Germany, September 15-17, 2008, Proceedings
2176:
asserts that the first can be cut into polygonal pieces which can be reassembled to form the second polygon.
714:
622:
3054:), the centroids of the vertices and of the solid shape are the same, but, in general, this is not true for
5016:{\displaystyle \lim _{n\to +\infty }R^{2}\cdot {\frac {n}{2}}\cdot \sin {\frac {2\pi }{n}}=\pi \cdot R^{2}}
585:
3799:
The simplest polygon such that it is not known if the regular form can be constructed with neusis or not.
5987:
5967:
4301:
4182:
4422:
is large, this approaches one half. Or, each vertex inside the square mesh connects four edges (lines).
3354:
are used in any dimension, with the distinction between the two that a polytope is necessarily bounded.)
5962:
5919:
5894:
5301:
4373:
2173:
1536:
3637:
The simplest polygon such that the regular form cannot be constructed with compass, straightedge, and
4436:
3258:
3249:
is a circuit of arcs of great circles (sides) and vertices on the surface of a sphere. It allows the
2133:
628:
17:
2099:{\displaystyle {\begin{aligned}A={\frac {1}{2}}(a_{1}\\{}+a_{2}\\{}+\cdots +a_{n-2}).\end{aligned}}}
1487:
328:: a polygon which self-intersects in a regular way. A polygon cannot be both a star and star-shaped.
6428:
6022:
4592:
4329:
4246:, where the angles between the sides depend on the type of mineral from which the crystal is made.
3578:
3305:
2128:
1441:
31:
2112:
If the polygon can be drawn on an equally spaced grid such that all its vertices are grid points,
6398:
5947:
5733:
4322:
4204:
4000:
3010:{\displaystyle C_{y}={\frac {1}{6A}}\sum _{i=0}^{n-1}(y_{i}+y_{i+1})(x_{i}y_{i+1}-x_{i+1}y_{i}).}
2838:{\displaystyle C_{x}={\frac {1}{6A}}\sum _{i=0}^{n-1}(x_{i}+x_{i+1})(x_{i}y_{i+1}-x_{i+1}y_{i}),}
3241:
The idea of a polygon has been generalized in various ways. Some of the more important include:
801:
5972:
5857:
5307:
4430:
2191:-gons with a given perimeter, the one with the largest area is regular (and therefore cyclic).
423:: the polygon's sides meet at right angles, i.e. all its interior angles are 90 or 270 degrees.
102:
5661:
5635:
5574:
5402:
5034:
4640:
4635:
A new universal etymological technological, and pronouncing dictionary of the
English language
2441:
6942:
6935:
6928:
6373:
6313:
5952:
5612:
5597:
5551:
5528:
5386:
5229:
4544:
4509:
3952:
3894:
3720:
3574:
3328:
1408:
434:
6467:
6445:
6433:
4909:
6599:
6546:
6257:
6027:
5957:
5899:
5836:
5152:
4875:
2294:
5323:
8:
6954:
6853:
6603:
6363:
6338:
6308:
6303:
6262:
5977:
5657:
4369:
4266:
4258:
4232:
3772:
3582:
3485:
Not generally recognised as a polygon in the
Euclidean plane, although it can exist as a
769:
420:
344:
338:
291:
91:
4900:(R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979: 147.
6823:
6773:
6723:
6680:
6650:
6610:
6573:
6391:
6368:
5909:
5140:
4879:
4853:
4765:
4559:
4197:
3294:
3023:
465:
370:
4805:
4485:
lies inside a simple polygon given by a sequence of line segments. This is called the
4372:
used in modelling and rendering. They are defined in a database, containing arrays of
3538:
The simplest polygon which can exist as a regular star. A star pentagon is known as a
6987:
6962:
6348:
5942:
5850:
5789:
5667:
5640:
5617:
5579:
5556:
5533:
5510:
5472:
5449:
5423:
5416:
5235:
5075:
4679:
4549:
4377:
4365:
4190:
3504:
3486:
3324:
3262:
3246:
2113:
793:
381:
120:
5824:
5067:
3910:
282:: the boundary of the polygon does not cross itself. All convex polygons are simple.
6966:
6531:
6520:
6509:
6498:
6489:
6480:
6419:
6415:
5877:
5766:
5362:
5279:
5275:
5130:
4883:
4863:
4757:
4529:
4486:
4262:
4236:
4216:
4212:
3556:
3391:
3317:
3301:
2211:
1553:
994:
426:
374:
310:
58:
5812:
5135:
4773:
6556:
6541:
6343:
6323:
6318:
6288:
6007:
5982:
5914:
5770:
5765:. Lecture Notes in Computer Science. Vol. 5193. Springer. pp. 744–755.
5694:
5467:
5148:
4871:
4633:
4519:
4166:
3992:
3969:
3934:
Used as an example in some philosophical discussions, for example in
Descartes's
3638:
3429:
2646:
2200:
1004:
825:
In this section, the vertices of the polygon under consideration are taken to be
777:
392:
350:
285:
187:
110:
4842:(2005). "The area of cyclic polygons: recent progress on Robbins' conjectures".
621:
were first studied by
Poinsot, in the same paper in which he describes the four
288:: Non-convex and simple. There is at least one interior angle greater than 180°.
6906:
6353:
6333:
6298:
6293:
5924:
5904:
5260:
4652:
4514:
4504:
4196:
The first known systematic study of non-convex polygons in general was made by
3996:
3387:
3375:
3287:
3273:
2187:-gons with given side lengths, the one with the largest area is cyclic. Of all
1590:
1548:
1000:
805:
764:
511:
481:
396:
385:
356:
319:
279:
270:
218:
147:
5351:"Nominalism and constructivism in seventeenth-century mathematical philosophy"
5035:"Slaying a geometrical 'Monster': finding the area of a crossed Quadrilateral"
4867:
6981:
6923:
6811:
6804:
6797:
6761:
6754:
6747:
6711:
6704:
6328:
6179:
6072:
5992:
5934:
5792:
5367:
5350:
4806:"Dergiades, Nikolaos, "An elementary proof of the isoperimetric inequality",
4794:. translators: J Massalski and C Mills Jr. D C Heath and Company: Boston, MA.
4569:
3914:
3780:
3522:
3512:
3415:
3309:
3272:
does not lie in a flat plane, but zigzags in three (or more) dimensions. The
315:
3290:
is an infinite sequence of sides and angles that do not lie in a flat plane.
171:. In contexts where one is concerned only with simple and solid polygons, a
6863:
6358:
6228:
6184:
6148:
6138:
6133:
4564:
4554:
4539:
4388:
4174:
3804:
3771:
The simplest polygon such that the regular form cannot be constructed with
3711:
3685:
3672:
3432:
3269:
618:
407:
364:
325:
266:
Polygons may be characterized by their convexity or type of non-convexity:
191:
179:
98:
4227:
6872:
6833:
6783:
6733:
6690:
6660:
6592:
6578:
6267:
6174:
6153:
6143:
4661:. Pure and Applied Mathematics. Vol. 61. Academic Press. p. 37.
4208:
3762:
3727:
3698:
3313:
3254:
1195:
4242:
Polygons appear in rock formations, most commonly as the flat facets of
817:
6858:
6842:
6792:
6742:
6699:
6669:
6583:
6272:
6128:
6118:
6002:
5445:
The universal book of mathematics: from Abracadabra to Zeno's paradoxes
5144:
4769:
4406:
squared triangles since there are two triangles in a square. There are
4336: in this section. Unsourced material may be challenged and removed.
4270:
3918:
3659:
3628:
3368:
3339:
1568:
of a simple polygon can also be computed if the lengths of the sides,
446:
6914:
6828:
6778:
6728:
6685:
6655:
6624:
6247:
6237:
6214:
6204:
6194:
6123:
6032:
5997:
5831:
Comparison of the different algorithms for Polygon Boolean operations
5797:
4858:
4281:
4170:
3977:
3944:
3901:
3646:
3539:
3439:
3406:
3280:
946:{\displaystyle (x_{0},y_{0}),(x_{1},y_{1}),\ldots ,(x_{n-1},y_{n-1})}
789:
5109:
Discrete and computational geometry: the Goodman-Pollack Festschrift
4761:
4311:
39:
6888:
6643:
6639:
6566:
6252:
6242:
6199:
6158:
6087:
6077:
6067:
5886:
5121:
Hass, Joel; Morgan, Frank (1996). "Geodesic nets on the 2-sphere".
4839:
4534:
3926:
3750:
3564:
3530:
3494:
3435:
3411:
3400:
3343:
3062:
3044:
1535:
The signed area depends on the ordering of the vertices and of the
198:
140:
48:
5842:
4207:
generalized the idea of polygons to the complex plane, where each
6897:
6867:
6634:
6629:
6620:
6561:
6209:
6189:
6102:
6097:
6092:
6082:
6057:
6012:
5827:, solutions to mathematical problems computing 2D and 3D polygons
4524:
4289:
4277:
4243:
3959:
3886:
3616:
3602:
3590:
3547:
3503:
The simplest polygon which can exist in the Euclidean plane. Can
3464:
3419:
2275:
4921:
A regular polygon with an infinite number of sides is a circle:
2649:
can be defined in two different ways, giving different answers:
245:
6837:
6787:
6737:
6694:
6664:
6615:
6551:
6017:
4254:
4178:
500:
360:
4675:
Beyond measure: a guided tour through nature, myth, and number
2206:
The area of a regular polygon is given in terms of the radius
6062:
5107:
Grunbaum, B.; "Are your polyhedra the same as my polyhedra",
4186:
3611:"Nonagon" mixes Latin with Greek; "enneagon" is pure Greek.
3477:
3250:
94:
3999:
for clarity of concatenated prefix numbers in the naming of
6587:
4606:
Grünbaum, B.; Are your polyhedra the same as my polyhedra?
4433:, it is often necessary to determine whether a given point
4250:
3226:{\displaystyle c_{y}={\frac {1}{n}}\sum _{i=0}^{n-1}y_{i}.}
3145:{\displaystyle c_{x}={\frac {1}{n}}\sum _{i=0}^{n-1}x_{i},}
175:
may refer only to a simple polygon or to a solid polygon.
5688:
Cratere with the blinding of Polyphemus and a naval battle
4896:
Chakerian, G. D. "A Distorted View of Geometry." Ch. 7 in
4608:
Discrete and comput. geom: the Goodman-Pollack festschrift
258:
Polygons are primarily classified by the number of sides.
79:
67:
5613:
On Understanding Understanding: A Philosophy of Knowledge
5555:, Continuum International Publishing Group, 2010, p. 26,
5234:. Springer Science & Business Media. pp. 88–90.
4292:, and the sides and base of each cell are also polygons.
4285:
953:
in order. For convenience in some formulas, the notation
197:
A polygon is a 2-dimensional example of the more general
76:
73:
3921:, have used the chiliagon as an example in discussions.
229:) 'corner' or 'angle'. It has been suggested that γόνυ (
186:. Some sources also consider closed polygonal chains in
108:
The segments of a closed polygonal chain are called its
5833:, compares capabilities, speed and numerical robustness
5748:"direct3d rendering, based on vertices & triangles"
4610:, ed. Aronov et al. Springer (2003) pp. 461–488. (
3573:
The simplest polygon such that the regular form is not
496:
300:: the boundary of the polygon crosses itself. The term
194:), even when the chain does not lie in a single plane.
3308:
analogous to an ordinary polygon, which exists in the
2235:
1282:
719:
670:
633:
596:
553:
5819:
How to draw monochrome orthogonal polygons on screens
5303:
The New Elements of Mathematics: Algebra and Geometry
4927:
4439:
3159:
3078:
3026:
2852:
2680:
2470:
2444:
2306:
2227:
2136:
1623:
1490:
1444:
1411:
1206:
1016:
831:
717:
668:
631:
588:
540:
82:
308:, but this usage risks confusion with the idea of a
118:. The points where two edges meet are the polygon's
70:
64:
5825:
comp.graphics.algorithms Frequently Asked Questions
5813:
Polygons, types of polygons, and polygon properties
4596:, Methuen and Co., 1948 (3rd Edition, Dover, 1973).
4177:), appearing as early as the 7th century B.C. on a
61:
5415:
5223:
5221:
5219:
5217:
5215:
5213:
5211:
5209:
5207:
5205:
5203:
5201:
5199:
5197:
5195:
5015:
4477:
4165:Polygons have been known since ancient times. The
3225:
3144:
3032:
3009:
2837:
2629:
2453:
2434:-gon inscribed in a unit-radius circle, with side
2419:
2263:
2161:
2098:
1525:
1476:
1430:
1394:
1183:
999:If the polygon is non-self-intersecting (that is,
945:
752:
703:
646:
609:
575:{\displaystyle \left(1-{\tfrac {2}{n}}\right)\pi }
574:
178:A polygonal chain may cross over itself, creating
5837:Interior angle sum of polygons: a general formula
5616:, 2nd ed, Fordham University Press, 1993, p. 86,
5193:
5191:
5189:
5187:
5185:
5183:
5181:
5179:
5177:
5175:
4387:Any surface is modelled as a tessellation called
3276:of the regular polytopes are well known examples.
399:. The polygon is also equilateral and tangential.
201:in any number of dimensions. There are many more
6979:
5787:
5422:(Online-Ausg. ed.). New York: McGraw-Hill.
5123:Proceedings of the American Mathematical Society
4929:
4710:"Calculating The Area And Centroid Of A Polygon"
2264:{\displaystyle A={\tfrac {1}{2}}\cdot p\cdot r.}
2199:Many specialized formulas apply to the areas of
437:to L intersects the polygon not more than twice.
4249:Regular hexagons can occur when the cooling of
3985:A degenerate polygon of infinitely many sides.
2172:For any two simple polygons of equal area, the
2109:The formula was described by Lopshits in 1963.
772:to the interior angle. Tracing around a convex
5172:
3641:. However, it can be constructed with neusis.
6399:
5858:
5719:Shephard, G.C.; "Regular complex polytopes",
5111:, Ed. Aronov et al., Springer (2003), p. 464.
4789:
485:– The sum of the interior angles of a simple
388:. The polygon is also cyclic and equiangular.
261:
5639:, reprint edition, Routledge, 2004, p. 202,
5261:https://dx.doi.org/10.1017/S0305004113000753
5094:
5092:
5090:
5088:
3020:In these formulas, the signed value of area
2461:can also be expressed trigonometrically as:
5666:. Courier Dover Publications. p. 162.
5255:
5253:
5251:
5032:
4825:Robbins, "Polygons inscribed in a circle",
4740:
4169:were known to the ancient Greeks, with the
522:sides ) can be considered to be made up of
6406:
6392:
5865:
5851:
5503:Merrill, John Calhoun and Odell, S. Jack,
5270:
5268:
5120:
5070:(1995). "Lectures on Polytopes". Springer
4658:Noneuclidean tesselations and their groups
3447:Polygon names and miscellaneous properties
441:
5578:, Oxford University Press, 2006, p. 124,
5529:An Introduction to Philosophical Analysis
5413:
5366:
5318:
5316:
5297:
5295:
5293:
5291:
5289:
5287:
5134:
5085:
4857:
4352:Learn how and when to remove this message
4253:forms areas of tightly packed columns of
3581:. However, it can be constructed using a
155:. The interior of a solid polygon is its
5663:A History of Greek Mathematics, Volume 1
5471:, 2nd ed, Addison-Wesley, 1999. p. 505.
5383:Kant's Metaphysics and Theory of Science
5348:
5342:
5248:
4792:Computation of areas of oriented figures
4671:
4226:
4156:
816:
704:{\displaystyle {\tfrac {\pi (p-2q)}{p}}}
617:degrees. The interior angles of regular
445:
332:
244:
38:
6971:List of regular polytopes and compounds
5760:
5601:, Sadlier and Co., Boston, 1856, p. 27.
5494:, Loyola University Press, 1928, p. 18.
5448:, John Wiley & Sons, 2004. p. 249.
5436:
5265:
5227:
4783:
2297:can be expressed trigonometrically as:
753:{\displaystyle {\tfrac {180(p-2q)}{p}}}
14:
6980:
5313:
5284:
4910:Area of a regular polygon – derivation
4707:
4651:
4398:points (vertices) per side, there are
2662:it is treated as two simple triangles.
5846:
5788:
5656:
5399:The Philosophical Works of David Hume
5033:De Villiers, Michael (January 2015).
4631:
4008:
610:{\displaystyle 180-{\tfrac {360}{n}}}
27:Plane figure bounded by line segments
5710:, 3rd Edn, Dover (pbk), 1973, p. 114
5330:. The Math Forum – Drexel University
5114:
4334:adding citations to reliable sources
4305:
4295:
3382:), noun use of neuter of πολύγωνος (
3065:of the vertex set of a polygon with
2640:
821:Coordinates of a non-convex pentagon
202:
5872:
4838:
4161:Historical image of polygons (1699)
3793:icosipentagon (or icosikaipentagon)
3346:. (In other conventions, the words
2194:
353:: both equilateral and equiangular.
347:: all edges are of the same length.
314:as one which exists in the complex
24:
5532:, 4th ed, Routledge, 1997, p. 56,
5401:, Volume 1, Black and Tait, 1826,
4942:
3846:heptacontagon (or hebdomecontagon)
3816:tetracontagon (or tessaracontagon)
3236:
988:
384:: all corners lie within the same
253:
25:
7004:
5781:
5723:Series 3 Volume 2, 1952, pp 82–97
5042:Learning and Teaching Mathematics
4678:. World Scientific. p. 258.
4638:. Oxford University. p. 404.
4402:squared squares in the mesh, or 2
4003:, though not all sources use it.
3740:enneadecagon (or enneakaidecagon)
792:and 0° for an angular "eight" or
304:is sometimes used in contrast to
240:
5468:College Algebra and Trigonometry
4310:
4173:, a non-convex regular polygon (
3866:enneacontagon (or enenecontagon)
3826:pentacontagon (or pentecontagon)
2119:In every polygon with perimeter
1438:is the squared distance between
414:
395:: all sides lie within the same
205:defined for different purposes.
57:
5809:, with Greek Numerical Prefixes
5754:
5740:
5726:
5713:
5700:
5681:
5650:
5627:
5604:
5589:
5566:
5552:Key Terms in Philosophy of Mind
5543:
5520:
5497:
5482:
5459:
5407:
5391:
5385:, Manchester University Press,
5375:
5324:"Naming Polygons and Polyhedra"
5159:
5101:
5061:
5052:
5026:
4915:
4903:
4890:
4845:Advances in Applied Mathematics
4832:
4750:The College Mathematics Journal
4581:
4478:{\displaystyle P=(x_{0},y_{0})}
4321:needs additional citations for
4211:dimension is accompanied by an
3937:Meditations on First Philosophy
3856:octacontagon (or ogdoëcontagon)
2274:This radius is also termed its
2162:{\displaystyle p^{2}>4\pi A}
1121:
647:{\displaystyle {\tfrac {p}{q}}}
249:Some different types of polygon
5598:Fundamental Philosophy, Vol II
5280:10.1080/00029890.2002.11919848
4936:
4819:
4798:
4734:
4701:
4692:
4665:
4645:
4625:
4603:, CUP hbk (1997), pbk. (1999).
4500:Boolean operations on polygons
4472:
4446:
4112:heptaconta- (or hebdomeconta-)
4070:tetraconta- (or tessaraconta-)
3836:hexacontagon (or hexecontagon)
3001:
2943:
2940:
2908:
2829:
2771:
2768:
2736:
2086:
2083:
2080:
2061:
2036:
2002:
1999:
1961:
1927:
1901:
1879:
1866:
1847:
1825:
1822:
1771:
1737:
1711:
1689:
1676:
1657:
1644:
1551:. This is commonly called the
1526:{\displaystyle (x_{j},y_{j}).}
1517:
1491:
1471:
1445:
1118:
1060:
940:
902:
890:
864:
858:
832:
740:
725:
691:
676:
373:: all sides are tangent to an
359:: all corners lie on a single
341:: all corner angles are equal.
233:) 'knee' may be the origin of
13:
1:
5734:"opengl vertex specification"
5636:History of Western Philosophy
5575:The Rise of Modern Philosophy
5169:, Dover Edition (1973), p. 4.
5136:10.1090/S0002-9939-96-03492-2
5072:Graduate Texts in Mathematics
4827:American Mathematical Monthly
4743:"The Surveyor's Area Formula"
4576:
4418:vertices per triangle. Where
3069:vertices has the coordinates
1477:{\displaystyle (x_{i},y_{i})}
514:. This is because any simple
429:with respect to a given line
5815:, with interactive animation
5771:10.1007/978-3-540-87744-8_62
5231:The Computer Graphics Manual
4222:
4140:enneaconta- (or eneneconta-)
4084:pentaconta- (or penteconta-)
2289:-gon in terms of the radius
2278:and is often represented as
768:– The exterior angle is the
225:) 'much', 'many' and γωνία (
208:
7:
4492:
4302:Polygon (computer graphics)
4257:, which may be seen at the
4126:octaconta- (or ogdoëconta-)
2666:
2183:determine the area. Of all
203:generalizations of polygons
190:to be a type of polygon (a
10:
7009:
6960:
6387:
4708:Bourke, Paul (July 1988).
4299:
4152:
4098:hexaconta- (or hexeconta-)
4042:icosi- (icosa- when alone)
3876:hectogon (or hecatontagon)
992:
662:), each interior angle is
658:-gon with central density
262:Convexity and intersection
184:self-intersecting polygons
29:
6281:
6227:
6167:
6111:
6050:
6041:
5933:
5885:
5506:Philosophy and Journalism
5488:McCormick, John Francis,
5381:Gottfried Martin (1955),
4912:from Math Open Reference.
4868:10.1016/j.aam.2004.08.006
4429:In computer graphics and
4280:, the surface of the wax
4056:triaconta- (or triconta-)
4033:
4024:
4016:
3358:
2647:self-intersecting polygon
788:of 360°, e.g. 720° for a
471:
6993:Euclidean plane geometry
5658:Heath, Sir Thomas Little
5509:, Longman, 1983, p. 47,
5414:Gibilisco, Stan (2003).
5368:10.1016/j.hm.2003.09.002
5349:Sepkoski, David (2005).
4618:
4376:(the coordinates of the
3579:compass and straightedge
2454:{\displaystyle \alpha ,}
2129:isoperimetric inequality
318:plane consisting of two
32:Polygon (disambiguation)
5721:Proc. London Math. Soc.
5228:Salomon, David (2011).
4391:. If a square mesh has
4205:Geoffrey Colin Shephard
3909:Philosophers including
1431:{\displaystyle Q_{i,j}}
812:
468:is assumed throughout.
442:Properties and formulas
5491:Scholastic Metaphysics
5308:Charles Sanders Peirce
5017:
4790:A.M. Lopshits (1963).
4672:Kappraff, Jay (2002).
4479:
4431:computational geometry
4239:
4162:
4001:quasiregular polyhedra
3259:Wythoff's construction
3227:
3209:
3146:
3128:
3034:
3011:
2907:
2839:
2735:
2631:
2455:
2430:The area of a regular
2421:
2285:The area of a regular
2265:
2174:Bolyai–Gerwien theorem
2163:
2100:
1543:-axis to the positive
1527:
1478:
1432:
1396:
1276:
1249:
1185:
1059:
947:
822:
754:
705:
648:
623:regular star polyhedra
611:
576:
462:
250:
139:sides; for example, a
103:closed polygonal chain
44:
5058:Coxeter (3rd Ed 1973)
5018:
4545:Polygon triangulation
4510:Constructible polygon
4480:
4230:
4200:in the 14th century.
4160:
3953:Constructible polygon
3895:Constructible polygon
3783:(or icosikaitetragon)
3721:Constructible polygon
3438:is also known as the
3329:partially ordered set
3228:
3183:
3147:
3102:
3035:
3012:
2881:
2840:
2709:
2632:
2456:
2422:
2266:
2164:
2101:
1528:
1479:
1433:
1397:
1250:
1223:
1186:
1033:
993:Further information:
948:
820:
755:
706:
649:
612:
577:
449:
333:Equality and symmetry
248:
42:
6098:Nonagon/Enneagon (9)
6028:Tangential trapezoid
5821:, by Herbert Glarner
5610:Potter, Vincent G.,
5418:Geometry demystified
5355:Historia Mathematica
4925:
4829:102, June–July 1995.
4808:Forum Mathematicorum
4741:Bart Braden (1986).
4722:on 16 September 2012
4632:Craig, John (1849).
4437:
4378:geometrical vertices
4330:improve this article
3714:(or heptakaidecagon)
3688:(or pentakaidecagon)
3675:(or tetrakaidecagon)
3257:(map making) and in
3157:
3076:
3024:
2850:
2678:
2468:
2442:
2304:
2295:circumscribed circle
2225:
2134:
1621:
1488:
1442:
1409:
1204:
1014:
829:
715:
666:
629:
586:
538:
101:connected to form a
30:For other uses, see
6955:pentagonal polytope
6854:Uniform 10-polytope
6414:Fundamental convex
6210:Megagon (1,000,000)
5978:Isosceles trapezoid
5807:What Are Polyhedra?
5633:Russell, Bertrand,
5442:Darling, David J.,
3995:, and advocated by
3765:(or icosikaitrigon)
3730:(or octakaidecagon)
3701:(or hexakaidecagon)
3662:(or triskaidecagon)
3583:neusis construction
3449:
3384:polygōnos/polugōnos
3380:polygōnon/polugōnon
2438:and interior angle
985:will also be used.
770:supplementary angle
6824:Uniform 9-polytope
6774:Uniform 8-polytope
6724:Uniform 7-polytope
6681:Uniform 6-polytope
6651:Uniform 5-polytope
6611:Uniform polychoron
6574:Uniform polyhedron
6422:in dimensions 2–10
6180:Icositetragon (24)
5790:Weisstein, Eric W.
5693:2013-11-12 at the
5465:Dugopolski, Mark,
5013:
4946:
4898:Mathematical Plums
4560:Synthetic geometry
4475:
4240:
4198:Thomas Bradwardine
4163:
3445:
3295:polygon with holes
3223:
3142:
3030:
3007:
2835:
2627:
2451:
2417:
2261:
2244:
2214:and its perimeter
2159:
2096:
2094:
1559:surveyor's formula
1523:
1474:
1428:
1392:
1383:
1181:
943:
823:
750:
748:
701:
699:
644:
642:
607:
605:
572:
562:
466:Euclidean geometry
463:
251:
159:, also known as a
135:is a polygon with
45:
6976:
6975:
6963:Polytope families
6420:uniform polytopes
6382:
6381:
6223:
6222:
6200:Myriagon (10,000)
6185:Triacontagon (30)
6149:Heptadecagon (17)
6139:Pentadecagon (15)
6134:Tetradecagon (14)
6073:Quadrilateral (4)
5943:Antiparallelogram
5708:Regular Polytopes
5706:Coxeter, H.S.M.;
5673:978-0-486-24073-2
5429:978-0-07-141650-4
5241:978-0-85729-886-7
5167:Regular polytopes
5165:Coxeter, H.S.M.;
5129:(12): 3843–3850.
5080:978-0-387-94365-7
4992:
4968:
4928:
4810:2, 2002, 129–130"
4685:978-981-02-4702-7
4641:Extract of p. 404
4593:Regular Polytopes
4550:Precision polygon
4368:, a polygon is a
4366:computer graphics
4362:
4361:
4354:
4296:Computer graphics
4191:Capitoline Museum
4150:
4149:
3989:
3988:
3487:spherical polygon
3263:uniform polyhedra
3247:spherical polygon
3181:
3100:
3033:{\displaystyle A}
2879:
2707:
2641:Self-intersecting
2622:
2595:
2562:
2538:
2513:
2497:
2415:
2396:
2358:
2334:
2243:
1642:
1614:are known, from:
1153:
1125:
1031:
794:antiparallelogram
747:
698:
641:
604:
561:
382:vertex-transitive
298:Self-intersecting
221:adjective πολύς (
217:derives from the
16:(Redirected from
7000:
6967:Regular polytope
6528:
6517:
6506:
6465:
6408:
6401:
6394:
6385:
6384:
6195:Chiliagon (1000)
6175:Icositrigon (23)
6154:Octadecagon (18)
6144:Hexadecagon (16)
6048:
6047:
5867:
5860:
5853:
5844:
5843:
5803:
5802:
5775:
5774:
5758:
5752:
5751:
5744:
5738:
5737:
5730:
5724:
5717:
5711:
5704:
5698:
5685:
5679:
5677:
5654:
5648:
5631:
5625:
5608:
5602:
5593:
5587:
5572:Kenny, Anthony,
5570:
5564:
5547:
5541:
5524:
5518:
5501:
5495:
5486:
5480:
5463:
5457:
5440:
5434:
5433:
5421:
5411:
5405:
5395:
5389:
5379:
5373:
5372:
5370:
5346:
5340:
5339:
5337:
5335:
5320:
5311:
5299:
5282:
5272:
5263:
5257:
5246:
5245:
5225:
5170:
5163:
5157:
5156:
5138:
5118:
5112:
5105:
5099:
5096:
5083:
5065:
5059:
5056:
5050:
5049:
5039:
5030:
5024:
5022:
5020:
5019:
5014:
5012:
5011:
4993:
4988:
4980:
4969:
4961:
4956:
4955:
4945:
4919:
4913:
4907:
4901:
4894:
4888:
4887:
4861:
4836:
4830:
4823:
4817:
4816:
4814:
4802:
4796:
4795:
4787:
4781:
4780:
4778:
4772:. Archived from
4747:
4738:
4732:
4731:
4729:
4727:
4721:
4715:. Archived from
4714:
4705:
4699:
4696:
4690:
4689:
4669:
4663:
4662:
4649:
4643:
4639:
4629:
4530:List of polygons
4487:point in polygon
4484:
4482:
4481:
4476:
4471:
4470:
4458:
4457:
4417:
4397:
4357:
4350:
4346:
4343:
4337:
4314:
4306:
4267:Devil's Postpile
4263:Northern Ireland
4259:Giant's Causeway
4237:Northern Ireland
4233:Giant's Causeway
4217:complex polygons
4167:regular polygons
4006:
4005:
3970:regular polygons
3450:
3444:
3422:are exceptions.
3394:with the suffix
3392:numerical prefix
3327:is an algebraic
3325:abstract polygon
3232:
3230:
3229:
3224:
3219:
3218:
3208:
3197:
3182:
3174:
3169:
3168:
3151:
3149:
3148:
3143:
3138:
3137:
3127:
3116:
3101:
3093:
3088:
3087:
3068:
3060:
3053:
3039:
3037:
3036:
3031:
3016:
3014:
3013:
3008:
3000:
2999:
2990:
2989:
2971:
2970:
2955:
2954:
2939:
2938:
2920:
2919:
2906:
2895:
2880:
2878:
2867:
2862:
2861:
2844:
2842:
2841:
2836:
2828:
2827:
2818:
2817:
2799:
2798:
2783:
2782:
2767:
2766:
2748:
2747:
2734:
2723:
2708:
2706:
2695:
2690:
2689:
2636:
2634:
2633:
2628:
2623:
2621:
2607:
2596:
2594:
2580:
2563:
2561:
2547:
2539:
2534:
2533:
2532:
2519:
2514:
2506:
2498:
2493:
2492:
2491:
2478:
2460:
2458:
2457:
2452:
2426:
2424:
2423:
2418:
2416:
2408:
2397:
2389:
2372:
2371:
2359:
2354:
2346:
2335:
2327:
2322:
2321:
2270:
2268:
2267:
2262:
2245:
2236:
2212:inscribed circle
2201:regular polygons
2195:Regular polygons
2168:
2166:
2165:
2160:
2146:
2145:
2105:
2103:
2102:
2097:
2095:
2079:
2078:
2054:
2053:
2035:
2034:
2010:
1998:
1997:
1973:
1972:
1954:
1953:
1926:
1925:
1913:
1912:
1894:
1893:
1878:
1877:
1859:
1858:
1846:
1845:
1833:
1821:
1820:
1796:
1795:
1783:
1782:
1764:
1763:
1736:
1735:
1723:
1722:
1704:
1703:
1688:
1687:
1669:
1668:
1656:
1655:
1643:
1635:
1554:shoelace formula
1546:
1542:
1532:
1530:
1529:
1524:
1516:
1515:
1503:
1502:
1483:
1481:
1480:
1475:
1470:
1469:
1457:
1456:
1437:
1435:
1434:
1429:
1427:
1426:
1401:
1399:
1398:
1393:
1388:
1387:
1380:
1379:
1350:
1349:
1324:
1323:
1300:
1299:
1275:
1264:
1248:
1237:
1219:
1218:
1190:
1188:
1187:
1182:
1177:
1176:
1164:
1163:
1154:
1151:
1149:
1148:
1136:
1135:
1126:
1123:
1117:
1116:
1107:
1106:
1088:
1087:
1072:
1071:
1058:
1047:
1032:
1024:
995:Shoelace formula
984:
952:
950:
949:
944:
939:
938:
920:
919:
889:
888:
876:
875:
857:
856:
844:
843:
759:
757:
756:
751:
749:
743:
720:
710:
708:
707:
702:
700:
694:
671:
653:
651:
650:
645:
643:
634:
625:: for a regular
616:
614:
613:
608:
606:
597:
581:
579:
578:
573:
568:
564:
563:
554:
529:
510:
499:
460:
450:Partitioning an
375:inscribed circle
162:polygonal region
89:
88:
85:
84:
81:
78:
75:
72:
69:
66:
63:
21:
7008:
7007:
7003:
7002:
7001:
6999:
6998:
6997:
6978:
6977:
6946:
6939:
6932:
6815:
6808:
6801:
6765:
6758:
6751:
6715:
6708:
6542:Regular polygon
6535:
6526:
6519:
6515:
6508:
6504:
6495:
6486:
6479:
6475:
6463:
6457:
6453:
6441:
6423:
6412:
6383:
6378:
6277:
6231:
6219:
6163:
6129:Tridecagon (13)
6119:Hendecagon (11)
6107:
6043:
6037:
6008:Right trapezoid
5929:
5881:
5871:
5784:
5779:
5778:
5759:
5755:
5746:
5745:
5741:
5732:
5731:
5727:
5718:
5714:
5705:
5701:
5695:Wayback Machine
5686:
5682:
5674:
5655:
5651:
5632:
5628:
5609:
5605:
5595:Balmes, James,
5594:
5590:
5571:
5567:
5548:
5544:
5526:Hospers, John,
5525:
5521:
5502:
5498:
5487:
5483:
5464:
5460:
5441:
5437:
5430:
5412:
5408:
5396:
5392:
5380:
5376:
5347:
5343:
5333:
5331:
5322:
5321:
5314:
5300:
5285:
5273:
5266:
5258:
5249:
5242:
5226:
5173:
5164:
5160:
5119:
5115:
5106:
5102:
5097:
5086:
5066:
5062:
5057:
5053:
5037:
5031:
5027:
5007:
5003:
4981:
4979:
4960:
4951:
4947:
4932:
4926:
4923:
4922:
4920:
4916:
4908:
4904:
4895:
4891:
4837:
4833:
4824:
4820:
4812:
4804:
4803:
4799:
4788:
4784:
4776:
4762:10.2307/2686282
4745:
4739:
4735:
4725:
4723:
4719:
4712:
4706:
4702:
4697:
4693:
4686:
4670:
4666:
4653:Magnus, Wilhelm
4650:
4646:
4630:
4626:
4621:
4588:Coxeter, H.S.M.
4584:
4579:
4574:
4520:Geometric shape
4495:
4466:
4462:
4453:
4449:
4438:
4435:
4434:
4407:
4392:
4358:
4347:
4341:
4338:
4327:
4315:
4304:
4298:
4288:is an array of
4225:
4215:one, to create
4189:and now in the
4155:
3649:(or duodecagon)
3639:angle trisector
3374:(a noun), from
3361:
3302:complex polygon
3274:Petrie polygons
3239:
3237:Generalizations
3214:
3210:
3198:
3187:
3173:
3164:
3160:
3158:
3155:
3154:
3133:
3129:
3117:
3106:
3092:
3083:
3079:
3077:
3074:
3073:
3066:
3055:
3048:
3025:
3022:
3021:
2995:
2991:
2979:
2975:
2960:
2956:
2950:
2946:
2928:
2924:
2915:
2911:
2896:
2885:
2871:
2866:
2857:
2853:
2851:
2848:
2847:
2823:
2819:
2807:
2803:
2788:
2784:
2778:
2774:
2756:
2752:
2743:
2739:
2724:
2713:
2699:
2694:
2685:
2681:
2679:
2676:
2675:
2669:
2643:
2611:
2606:
2584:
2579:
2551:
2546:
2528:
2524:
2520:
2518:
2505:
2487:
2483:
2479:
2477:
2469:
2466:
2465:
2443:
2440:
2439:
2407:
2388:
2367:
2363:
2347:
2345:
2326:
2317:
2313:
2305:
2302:
2301:
2234:
2226:
2223:
2222:
2197:
2141:
2137:
2135:
2132:
2131:
2093:
2092:
2068:
2064:
2043:
2039:
2024:
2020:
2009:
2006:
2005:
1987:
1983:
1968:
1964:
1943:
1939:
1921:
1917:
1908:
1904:
1889:
1885:
1873:
1869:
1854:
1850:
1841:
1837:
1832:
1829:
1828:
1810:
1806:
1791:
1787:
1778:
1774:
1753:
1749:
1731:
1727:
1718:
1714:
1699:
1695:
1683:
1679:
1664:
1660:
1651:
1647:
1634:
1624:
1622:
1619:
1618:
1612:
1606:
1599:
1591:exterior angles
1587:
1581:
1574:
1544:
1540:
1511:
1507:
1498:
1494:
1489:
1486:
1485:
1465:
1461:
1452:
1448:
1443:
1440:
1439:
1416:
1412:
1410:
1407:
1406:
1382:
1381:
1357:
1353:
1351:
1333:
1329:
1326:
1325:
1307:
1303:
1301:
1289:
1285:
1278:
1277:
1265:
1254:
1238:
1227:
1214:
1210:
1205:
1202:
1201:
1172:
1168:
1159:
1155:
1152: and
1150:
1144:
1140:
1131:
1127:
1122:
1112:
1108:
1096:
1092:
1077:
1073:
1067:
1063:
1048:
1037:
1023:
1015:
1012:
1011:
997:
991:
989:Simple polygons
982:
975:
967:
960:
954:
928:
924:
909:
905:
884:
880:
871:
867:
852:
848:
839:
835:
830:
827:
826:
815:
808:of the polygon.
721:
718:
716:
713:
712:
672:
669:
667:
664:
663:
632:
630:
627:
626:
595:
587:
584:
583:
552:
545:
541:
539:
536:
535:
523:
504:
490:
474:
455:
444:
417:
393:edge-transitive
335:
311:complex polygon
264:
256:
254:Number of sides
243:
211:
188:Euclidean space
60:
56:
35:
28:
23:
22:
15:
12:
11:
5:
7006:
6996:
6995:
6990:
6974:
6973:
6958:
6957:
6948:
6944:
6937:
6930:
6926:
6917:
6900:
6891:
6880:
6879:
6877:
6875:
6870:
6861:
6856:
6850:
6849:
6847:
6845:
6840:
6831:
6826:
6820:
6819:
6817:
6813:
6806:
6799:
6795:
6790:
6781:
6776:
6770:
6769:
6767:
6763:
6756:
6749:
6745:
6740:
6731:
6726:
6720:
6719:
6717:
6713:
6706:
6702:
6697:
6688:
6683:
6677:
6676:
6674:
6672:
6667:
6658:
6653:
6647:
6646:
6637:
6632:
6627:
6618:
6613:
6607:
6606:
6597:
6595:
6590:
6581:
6576:
6570:
6569:
6564:
6559:
6554:
6549:
6544:
6538:
6537:
6533:
6529:
6524:
6513:
6502:
6493:
6484:
6477:
6471:
6461:
6455:
6449:
6443:
6437:
6431:
6425:
6424:
6413:
6411:
6410:
6403:
6396:
6388:
6380:
6379:
6377:
6376:
6371:
6366:
6361:
6356:
6351:
6346:
6341:
6336:
6334:Pseudotriangle
6331:
6326:
6321:
6316:
6311:
6306:
6301:
6296:
6291:
6285:
6283:
6279:
6278:
6276:
6275:
6270:
6265:
6260:
6255:
6250:
6245:
6240:
6234:
6232:
6225:
6224:
6221:
6220:
6218:
6217:
6212:
6207:
6202:
6197:
6192:
6187:
6182:
6177:
6171:
6169:
6165:
6164:
6162:
6161:
6156:
6151:
6146:
6141:
6136:
6131:
6126:
6124:Dodecagon (12)
6121:
6115:
6113:
6109:
6108:
6106:
6105:
6100:
6095:
6090:
6085:
6080:
6075:
6070:
6065:
6060:
6054:
6052:
6045:
6039:
6038:
6036:
6035:
6030:
6025:
6020:
6015:
6010:
6005:
6000:
5995:
5990:
5985:
5980:
5975:
5970:
5965:
5960:
5955:
5950:
5945:
5939:
5937:
5935:Quadrilaterals
5931:
5930:
5928:
5927:
5922:
5917:
5912:
5907:
5902:
5897:
5891:
5889:
5883:
5882:
5870:
5869:
5862:
5855:
5847:
5841:
5840:
5834:
5828:
5822:
5816:
5810:
5804:
5783:
5782:External links
5780:
5777:
5776:
5753:
5739:
5725:
5712:
5699:
5680:
5672:
5649:
5626:
5603:
5588:
5565:
5549:Mandik, Pete,
5542:
5519:
5496:
5481:
5458:
5435:
5428:
5406:
5390:
5374:
5341:
5312:
5283:
5264:
5247:
5240:
5171:
5158:
5113:
5100:
5084:
5068:Günter Ziegler
5060:
5051:
5025:
5010:
5006:
5002:
4999:
4996:
4991:
4987:
4984:
4978:
4975:
4972:
4967:
4964:
4959:
4954:
4950:
4944:
4941:
4938:
4935:
4931:
4914:
4902:
4889:
4852:(4): 690–696.
4831:
4818:
4797:
4782:
4779:on 2012-11-07.
4756:(4): 326–337.
4733:
4700:
4691:
4684:
4664:
4644:
4623:
4622:
4620:
4617:
4616:
4615:
4604:
4599:Cromwell, P.;
4597:
4583:
4580:
4578:
4575:
4573:
4572:
4567:
4562:
4557:
4552:
4547:
4542:
4537:
4532:
4527:
4522:
4517:
4515:Cyclic polygon
4512:
4507:
4505:Complete graph
4502:
4496:
4494:
4491:
4474:
4469:
4465:
4461:
4456:
4452:
4448:
4445:
4442:
4360:
4359:
4318:
4316:
4309:
4300:Main article:
4297:
4294:
4224:
4221:
4154:
4151:
4148:
4147:
4144:
4141:
4138:
4134:
4133:
4130:
4127:
4124:
4120:
4119:
4116:
4113:
4110:
4106:
4105:
4102:
4099:
4096:
4092:
4091:
4088:
4085:
4082:
4078:
4077:
4074:
4071:
4068:
4064:
4063:
4060:
4057:
4054:
4050:
4049:
4046:
4043:
4040:
4036:
4035:
4032:
4029:
4026:
4022:
4021:
4018:
4015:
4010:
3997:John H. Conway
3987:
3986:
3983:
3980:
3974:
3973:
3965:
3962:
3956:
3955:
3950:
3947:
3941:
3940:
3932:
3929:
3923:
3922:
3911:René Descartes
3907:
3904:
3898:
3897:
3892:
3889:
3883:
3882:
3880:
3877:
3873:
3872:
3870:
3867:
3863:
3862:
3860:
3857:
3853:
3852:
3850:
3847:
3843:
3842:
3840:
3837:
3833:
3832:
3830:
3827:
3823:
3822:
3820:
3817:
3813:
3812:
3810:
3807:
3801:
3800:
3797:
3794:
3790:
3789:
3787:
3784:
3777:
3776:
3769:
3766:
3759:
3758:
3756:
3753:
3747:
3746:
3744:
3741:
3737:
3736:
3734:
3731:
3724:
3723:
3718:
3715:
3708:
3707:
3705:
3702:
3695:
3694:
3692:
3689:
3682:
3681:
3679:
3676:
3669:
3668:
3666:
3663:
3656:
3655:
3653:
3650:
3643:
3642:
3635:
3632:
3631:(or undecagon)
3625:
3624:
3622:
3619:
3613:
3612:
3609:
3606:
3599:
3598:
3596:
3593:
3587:
3586:
3571:
3568:
3561:
3560:
3553:
3550:
3544:
3543:
3536:
3533:
3527:
3526:
3519:
3516:
3509:
3508:
3501:
3498:
3491:
3490:
3483:
3480:
3474:
3473:
3470:
3467:
3461:
3460:
3457:
3454:
3360:
3357:
3356:
3355:
3336:
3321:
3298:
3291:
3288:skew apeirogon
3284:
3277:
3266:
3238:
3235:
3234:
3233:
3222:
3217:
3213:
3207:
3204:
3201:
3196:
3193:
3190:
3186:
3180:
3177:
3172:
3167:
3163:
3152:
3141:
3136:
3132:
3126:
3123:
3120:
3115:
3112:
3109:
3105:
3099:
3096:
3091:
3086:
3082:
3040:must be used.
3029:
3018:
3017:
3006:
3003:
2998:
2994:
2988:
2985:
2982:
2978:
2974:
2969:
2966:
2963:
2959:
2953:
2949:
2945:
2942:
2937:
2934:
2931:
2927:
2923:
2918:
2914:
2910:
2905:
2902:
2899:
2894:
2891:
2888:
2884:
2877:
2874:
2870:
2865:
2860:
2856:
2845:
2834:
2831:
2826:
2822:
2816:
2813:
2810:
2806:
2802:
2797:
2794:
2791:
2787:
2781:
2777:
2773:
2770:
2765:
2762:
2759:
2755:
2751:
2746:
2742:
2738:
2733:
2730:
2727:
2722:
2719:
2716:
2712:
2705:
2702:
2698:
2693:
2688:
2684:
2668:
2665:
2664:
2663:
2659:
2645:The area of a
2642:
2639:
2638:
2637:
2626:
2620:
2617:
2614:
2610:
2605:
2602:
2599:
2593:
2590:
2587:
2583:
2578:
2575:
2572:
2569:
2566:
2560:
2557:
2554:
2550:
2545:
2542:
2537:
2531:
2527:
2523:
2517:
2512:
2509:
2504:
2501:
2496:
2490:
2486:
2482:
2476:
2473:
2450:
2447:
2428:
2427:
2414:
2411:
2406:
2403:
2400:
2395:
2392:
2387:
2384:
2381:
2378:
2375:
2370:
2366:
2362:
2357:
2353:
2350:
2344:
2341:
2338:
2333:
2330:
2325:
2320:
2316:
2312:
2309:
2272:
2271:
2260:
2257:
2254:
2251:
2248:
2242:
2239:
2233:
2230:
2196:
2193:
2158:
2155:
2152:
2149:
2144:
2140:
2114:Pick's theorem
2107:
2106:
2091:
2088:
2085:
2082:
2077:
2074:
2071:
2067:
2063:
2060:
2057:
2052:
2049:
2046:
2042:
2038:
2033:
2030:
2027:
2023:
2019:
2016:
2013:
2008:
2007:
2004:
2001:
1996:
1993:
1990:
1986:
1982:
1979:
1976:
1971:
1967:
1963:
1960:
1957:
1952:
1949:
1946:
1942:
1938:
1935:
1932:
1929:
1924:
1920:
1916:
1911:
1907:
1903:
1900:
1897:
1892:
1888:
1884:
1881:
1876:
1872:
1868:
1865:
1862:
1857:
1853:
1849:
1844:
1840:
1836:
1831:
1830:
1827:
1824:
1819:
1816:
1813:
1809:
1805:
1802:
1799:
1794:
1790:
1786:
1781:
1777:
1773:
1770:
1767:
1762:
1759:
1756:
1752:
1748:
1745:
1742:
1739:
1734:
1730:
1726:
1721:
1717:
1713:
1710:
1707:
1702:
1698:
1694:
1691:
1686:
1682:
1678:
1675:
1672:
1667:
1663:
1659:
1654:
1650:
1646:
1641:
1638:
1633:
1630:
1627:
1626:
1610:
1604:
1597:
1585:
1579:
1572:
1549:absolute value
1522:
1519:
1514:
1510:
1506:
1501:
1497:
1493:
1473:
1468:
1464:
1460:
1455:
1451:
1447:
1425:
1422:
1419:
1415:
1403:
1402:
1391:
1386:
1378:
1375:
1372:
1369:
1366:
1363:
1360:
1356:
1352:
1348:
1345:
1342:
1339:
1336:
1332:
1328:
1327:
1322:
1319:
1316:
1313:
1310:
1306:
1302:
1298:
1295:
1292:
1288:
1284:
1283:
1281:
1274:
1271:
1268:
1263:
1260:
1257:
1253:
1247:
1244:
1241:
1236:
1233:
1230:
1226:
1222:
1217:
1213:
1209:
1192:
1191:
1180:
1175:
1171:
1167:
1162:
1158:
1147:
1143:
1139:
1134:
1130:
1120:
1115:
1111:
1105:
1102:
1099:
1095:
1091:
1086:
1083:
1080:
1076:
1070:
1066:
1062:
1057:
1054:
1051:
1046:
1043:
1040:
1036:
1030:
1027:
1022:
1019:
1003:), the signed
990:
987:
980:
973:
965:
958:
942:
937:
934:
931:
927:
923:
918:
915:
912:
908:
904:
901:
898:
895:
892:
887:
883:
879:
874:
870:
866:
863:
860:
855:
851:
847:
842:
838:
834:
814:
811:
810:
809:
806:turning number
765:Exterior angle
761:
746:
742:
739:
736:
733:
730:
727:
724:
697:
693:
690:
687:
684:
681:
678:
675:
640:
637:
603:
600:
594:
591:
571:
567:
560:
557:
551:
548:
544:
518:-gon ( having
482:Interior angle
473:
470:
443:
440:
439:
438:
424:
416:
413:
401:
400:
397:symmetry orbit
389:
386:symmetry orbit
378:
368:
354:
348:
342:
334:
331:
330:
329:
323:
295:
289:
283:
277:
274:
263:
260:
255:
252:
242:
241:Classification
239:
210:
207:
168:polygonal area
148:simple polygon
26:
9:
6:
4:
3:
2:
7005:
6994:
6991:
6989:
6986:
6985:
6983:
6972:
6968:
6964:
6959:
6956:
6952:
6949:
6947:
6940:
6933:
6927:
6925:
6921:
6918:
6916:
6912:
6908:
6904:
6901:
6899:
6895:
6892:
6890:
6886:
6882:
6881:
6878:
6876:
6874:
6871:
6869:
6865:
6862:
6860:
6857:
6855:
6852:
6851:
6848:
6846:
6844:
6841:
6839:
6835:
6832:
6830:
6827:
6825:
6822:
6821:
6818:
6816:
6809:
6802:
6796:
6794:
6791:
6789:
6785:
6782:
6780:
6777:
6775:
6772:
6771:
6768:
6766:
6759:
6752:
6746:
6744:
6741:
6739:
6735:
6732:
6730:
6727:
6725:
6722:
6721:
6718:
6716:
6709:
6703:
6701:
6698:
6696:
6692:
6689:
6687:
6684:
6682:
6679:
6678:
6675:
6673:
6671:
6668:
6666:
6662:
6659:
6657:
6654:
6652:
6649:
6648:
6645:
6641:
6638:
6636:
6633:
6631:
6630:Demitesseract
6628:
6626:
6622:
6619:
6617:
6614:
6612:
6609:
6608:
6605:
6601:
6598:
6596:
6594:
6591:
6589:
6585:
6582:
6580:
6577:
6575:
6572:
6571:
6568:
6565:
6563:
6560:
6558:
6555:
6553:
6550:
6548:
6545:
6543:
6540:
6539:
6536:
6530:
6527:
6523:
6516:
6512:
6505:
6501:
6496:
6492:
6487:
6483:
6478:
6476:
6474:
6470:
6460:
6456:
6454:
6452:
6448:
6444:
6442:
6440:
6436:
6432:
6430:
6427:
6426:
6421:
6417:
6409:
6404:
6402:
6397:
6395:
6390:
6389:
6386:
6375:
6374:Weakly simple
6372:
6370:
6367:
6365:
6362:
6360:
6357:
6355:
6352:
6350:
6347:
6345:
6342:
6340:
6337:
6335:
6332:
6330:
6327:
6325:
6322:
6320:
6317:
6315:
6314:Infinite skew
6312:
6310:
6307:
6305:
6302:
6300:
6297:
6295:
6292:
6290:
6287:
6286:
6284:
6280:
6274:
6271:
6269:
6266:
6264:
6261:
6259:
6256:
6254:
6251:
6249:
6246:
6244:
6241:
6239:
6236:
6235:
6233:
6230:
6229:Star polygons
6226:
6216:
6215:Apeirogon (∞)
6213:
6211:
6208:
6206:
6203:
6201:
6198:
6196:
6193:
6191:
6188:
6186:
6183:
6181:
6178:
6176:
6173:
6172:
6170:
6166:
6160:
6159:Icosagon (20)
6157:
6155:
6152:
6150:
6147:
6145:
6142:
6140:
6137:
6135:
6132:
6130:
6127:
6125:
6122:
6120:
6117:
6116:
6114:
6110:
6104:
6101:
6099:
6096:
6094:
6091:
6089:
6086:
6084:
6081:
6079:
6076:
6074:
6071:
6069:
6066:
6064:
6061:
6059:
6056:
6055:
6053:
6049:
6046:
6040:
6034:
6031:
6029:
6026:
6024:
6021:
6019:
6016:
6014:
6011:
6009:
6006:
6004:
6001:
5999:
5996:
5994:
5993:Parallelogram
5991:
5989:
5988:Orthodiagonal
5986:
5984:
5981:
5979:
5976:
5974:
5971:
5969:
5968:Ex-tangential
5966:
5964:
5961:
5959:
5956:
5954:
5951:
5949:
5946:
5944:
5941:
5940:
5938:
5936:
5932:
5926:
5923:
5921:
5918:
5916:
5913:
5911:
5908:
5906:
5903:
5901:
5898:
5896:
5893:
5892:
5890:
5888:
5884:
5879:
5875:
5868:
5863:
5861:
5856:
5854:
5849:
5848:
5845:
5838:
5835:
5832:
5829:
5826:
5823:
5820:
5817:
5814:
5811:
5808:
5805:
5800:
5799:
5794:
5791:
5786:
5785:
5772:
5768:
5764:
5757:
5749:
5743:
5735:
5729:
5722:
5716:
5709:
5703:
5696:
5692:
5689:
5684:
5675:
5669:
5665:
5664:
5659:
5653:
5646:
5645:0-415-32505-6
5642:
5638:
5637:
5630:
5623:
5622:0-8232-1486-9
5619:
5615:
5614:
5607:
5600:
5599:
5592:
5585:
5584:0-19-875277-6
5581:
5577:
5576:
5569:
5562:
5561:1-84706-349-7
5558:
5554:
5553:
5546:
5539:
5538:0-415-15792-7
5535:
5531:
5530:
5523:
5516:
5515:0-582-28157-1
5512:
5508:
5507:
5500:
5493:
5492:
5485:
5478:
5477:0-201-34712-1
5474:
5470:
5469:
5462:
5455:
5454:0-471-27047-4
5451:
5447:
5446:
5439:
5431:
5425:
5420:
5419:
5410:
5404:
5400:
5394:
5388:
5384:
5378:
5369:
5364:
5360:
5356:
5352:
5345:
5329:
5325:
5319:
5317:
5310:(1976), p.298
5309:
5305:
5304:
5298:
5296:
5294:
5292:
5290:
5288:
5281:
5277:
5271:
5269:
5262:
5256:
5254:
5252:
5243:
5237:
5233:
5232:
5224:
5222:
5220:
5218:
5216:
5214:
5212:
5210:
5208:
5206:
5204:
5202:
5200:
5198:
5196:
5194:
5192:
5190:
5188:
5186:
5184:
5182:
5180:
5178:
5176:
5168:
5162:
5154:
5150:
5146:
5142:
5137:
5132:
5128:
5124:
5117:
5110:
5104:
5095:
5093:
5091:
5089:
5081:
5077:
5073:
5069:
5064:
5055:
5047:
5043:
5036:
5029:
5008:
5004:
5000:
4997:
4994:
4989:
4985:
4982:
4976:
4973:
4970:
4965:
4962:
4957:
4952:
4948:
4939:
4933:
4918:
4911:
4906:
4899:
4893:
4885:
4881:
4877:
4873:
4869:
4865:
4860:
4855:
4851:
4847:
4846:
4841:
4835:
4828:
4822:
4811:
4809:
4801:
4793:
4786:
4775:
4771:
4767:
4763:
4759:
4755:
4751:
4744:
4737:
4718:
4711:
4704:
4695:
4687:
4681:
4677:
4676:
4668:
4660:
4659:
4654:
4648:
4642:
4637:
4636:
4628:
4624:
4613:
4609:
4605:
4602:
4598:
4595:
4594:
4589:
4586:
4585:
4571:
4570:Tiling puzzle
4568:
4566:
4563:
4561:
4558:
4556:
4553:
4551:
4548:
4546:
4543:
4541:
4538:
4536:
4533:
4531:
4528:
4526:
4523:
4521:
4518:
4516:
4513:
4511:
4508:
4506:
4503:
4501:
4498:
4497:
4490:
4488:
4467:
4463:
4459:
4454:
4450:
4443:
4440:
4432:
4427:
4423:
4421:
4415:
4411:
4405:
4401:
4395:
4390:
4385:
4383:
4379:
4375:
4371:
4367:
4356:
4353:
4345:
4335:
4331:
4325:
4324:
4319:This section
4317:
4313:
4308:
4307:
4303:
4293:
4291:
4287:
4283:
4279:
4274:
4272:
4268:
4264:
4260:
4256:
4252:
4247:
4245:
4238:
4234:
4229:
4220:
4218:
4214:
4210:
4206:
4201:
4199:
4194:
4192:
4188:
4184:
4180:
4176:
4172:
4168:
4159:
4145:
4142:
4139:
4136:
4135:
4131:
4128:
4125:
4122:
4121:
4117:
4114:
4111:
4108:
4107:
4103:
4100:
4097:
4094:
4093:
4089:
4086:
4083:
4080:
4079:
4075:
4072:
4069:
4066:
4065:
4061:
4058:
4055:
4052:
4051:
4047:
4044:
4041:
4038:
4037:
4030:
4027:
4023:
4020:final suffix
4019:
4014:
4011:
4007:
4004:
4002:
3998:
3994:
3984:
3981:
3979:
3976:
3975:
3972:to a circle.
3971:
3966:
3963:
3961:
3958:
3957:
3954:
3951:
3948:
3946:
3943:
3942:
3939:
3938:
3933:
3930:
3928:
3925:
3924:
3920:
3916:
3915:Immanuel Kant
3912:
3908:
3905:
3903:
3900:
3899:
3896:
3893:
3890:
3888:
3885:
3884:
3881:
3878:
3875:
3874:
3871:
3868:
3865:
3864:
3861:
3858:
3855:
3854:
3851:
3848:
3845:
3844:
3841:
3838:
3835:
3834:
3831:
3828:
3825:
3824:
3821:
3818:
3815:
3814:
3811:
3808:
3806:
3803:
3802:
3798:
3795:
3792:
3791:
3788:
3785:
3782:
3781:icositetragon
3779:
3778:
3774:
3770:
3767:
3764:
3761:
3760:
3757:
3754:
3752:
3749:
3748:
3745:
3742:
3739:
3738:
3735:
3732:
3729:
3726:
3725:
3722:
3719:
3716:
3713:
3710:
3709:
3706:
3703:
3700:
3697:
3696:
3693:
3690:
3687:
3684:
3683:
3680:
3677:
3674:
3671:
3670:
3667:
3664:
3661:
3658:
3657:
3654:
3651:
3648:
3645:
3644:
3640:
3636:
3633:
3630:
3627:
3626:
3623:
3620:
3618:
3615:
3614:
3610:
3607:
3605:(or enneagon)
3604:
3601:
3600:
3597:
3594:
3592:
3589:
3588:
3584:
3580:
3576:
3575:constructible
3572:
3569:
3567:(or septagon)
3566:
3563:
3562:
3558:
3554:
3551:
3549:
3546:
3545:
3542:or pentacle.
3541:
3537:
3534:
3532:
3529:
3528:
3524:
3520:
3517:
3515:(or tetragon)
3514:
3513:quadrilateral
3511:
3510:
3506:
3502:
3499:
3496:
3493:
3492:
3488:
3484:
3481:
3479:
3476:
3475:
3471:
3468:
3466:
3463:
3462:
3458:
3455:
3452:
3451:
3448:
3443:
3441:
3437:
3434:
3431:
3426:
3423:
3421:
3417:
3416:quadrilateral
3413:
3409:
3408:
3403:
3402:
3397:
3393:
3389:
3385:
3381:
3377:
3373:
3370:
3366:
3353:
3349:
3345:
3341:
3337:
3334:
3330:
3326:
3322:
3319:
3315:
3311:
3310:complex plane
3307:
3306:configuration
3303:
3299:
3296:
3292:
3289:
3285:
3282:
3278:
3275:
3271:
3267:
3264:
3260:
3256:
3252:
3248:
3244:
3243:
3242:
3220:
3215:
3211:
3205:
3202:
3199:
3194:
3191:
3188:
3184:
3178:
3175:
3170:
3165:
3161:
3153:
3139:
3134:
3130:
3124:
3121:
3118:
3113:
3110:
3107:
3103:
3097:
3094:
3089:
3084:
3080:
3072:
3071:
3070:
3064:
3058:
3051:
3046:
3041:
3027:
3004:
2996:
2992:
2986:
2983:
2980:
2976:
2972:
2967:
2964:
2961:
2957:
2951:
2947:
2935:
2932:
2929:
2925:
2921:
2916:
2912:
2903:
2900:
2897:
2892:
2889:
2886:
2882:
2875:
2872:
2868:
2863:
2858:
2854:
2846:
2832:
2824:
2820:
2814:
2811:
2808:
2804:
2800:
2795:
2792:
2789:
2785:
2779:
2775:
2763:
2760:
2757:
2753:
2749:
2744:
2740:
2731:
2728:
2725:
2720:
2717:
2714:
2710:
2703:
2700:
2696:
2691:
2686:
2682:
2674:
2673:
2672:
2660:
2656:
2652:
2651:
2650:
2648:
2624:
2618:
2615:
2612:
2608:
2603:
2600:
2597:
2591:
2588:
2585:
2581:
2576:
2573:
2570:
2567:
2564:
2558:
2555:
2552:
2548:
2543:
2540:
2535:
2529:
2525:
2521:
2515:
2510:
2507:
2502:
2499:
2494:
2488:
2484:
2480:
2474:
2471:
2464:
2463:
2462:
2448:
2445:
2437:
2433:
2412:
2409:
2404:
2401:
2398:
2393:
2390:
2385:
2382:
2379:
2376:
2373:
2368:
2364:
2360:
2355:
2351:
2348:
2342:
2339:
2336:
2331:
2328:
2323:
2318:
2314:
2310:
2307:
2300:
2299:
2298:
2296:
2292:
2288:
2283:
2281:
2277:
2258:
2255:
2252:
2249:
2246:
2240:
2237:
2231:
2228:
2221:
2220:
2219:
2217:
2213:
2209:
2204:
2202:
2192:
2190:
2186:
2182:
2177:
2175:
2170:
2156:
2153:
2150:
2147:
2142:
2138:
2130:
2126:
2122:
2117:
2115:
2110:
2089:
2075:
2072:
2069:
2065:
2058:
2055:
2050:
2047:
2044:
2040:
2031:
2028:
2025:
2021:
2017:
2014:
2011:
1994:
1991:
1988:
1984:
1980:
1977:
1974:
1969:
1965:
1958:
1955:
1950:
1947:
1944:
1940:
1936:
1933:
1930:
1922:
1918:
1914:
1909:
1905:
1898:
1895:
1890:
1886:
1882:
1874:
1870:
1863:
1860:
1855:
1851:
1842:
1838:
1834:
1817:
1814:
1811:
1807:
1803:
1800:
1797:
1792:
1788:
1784:
1779:
1775:
1768:
1765:
1760:
1757:
1754:
1750:
1746:
1743:
1740:
1732:
1728:
1724:
1719:
1715:
1708:
1705:
1700:
1696:
1692:
1684:
1680:
1673:
1670:
1665:
1661:
1652:
1648:
1639:
1636:
1631:
1628:
1617:
1616:
1615:
1613:
1603:
1596:
1592:
1588:
1578:
1571:
1567:
1562:
1560:
1556:
1555:
1550:
1538:
1533:
1520:
1512:
1508:
1504:
1499:
1495:
1466:
1462:
1458:
1453:
1449:
1423:
1420:
1417:
1413:
1389:
1384:
1376:
1373:
1370:
1367:
1364:
1361:
1358:
1354:
1346:
1343:
1340:
1337:
1334:
1330:
1320:
1317:
1314:
1311:
1308:
1304:
1296:
1293:
1290:
1286:
1279:
1272:
1269:
1266:
1261:
1258:
1255:
1251:
1245:
1242:
1239:
1234:
1231:
1228:
1224:
1220:
1215:
1211:
1207:
1200:
1199:
1198:
1197:
1178:
1173:
1169:
1165:
1160:
1156:
1145:
1141:
1137:
1132:
1128:
1113:
1109:
1103:
1100:
1097:
1093:
1089:
1084:
1081:
1078:
1074:
1068:
1064:
1055:
1052:
1049:
1044:
1041:
1038:
1034:
1028:
1025:
1020:
1017:
1010:
1009:
1008:
1006:
1002:
996:
986:
979:
972:
968:
961:
935:
932:
929:
925:
921:
916:
913:
910:
906:
899:
896:
893:
885:
881:
877:
872:
868:
861:
853:
849:
845:
840:
836:
819:
807:
803:
799:
795:
791:
787:
783:
779:
775:
771:
767:
766:
762:
744:
737:
734:
731:
728:
722:
695:
688:
685:
682:
679:
673:
661:
657:
638:
635:
624:
620:
619:star polygons
601:
598:
592:
589:
569:
565:
558:
555:
549:
546:
542:
533:
527:
521:
517:
513:
508:
502:
498:
494:
488:
484:
483:
479:
478:
477:
469:
467:
458:
453:
448:
436:
433:: every line
432:
428:
425:
422:
419:
418:
415:Miscellaneous
412:
410:
409:
398:
394:
390:
387:
383:
379:
376:
372:
369:
366:
363:, called the
362:
358:
355:
352:
349:
346:
343:
340:
337:
336:
327:
324:
321:
317:
313:
312:
307:
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299:
296:
293:
290:
287:
284:
281:
278:
275:
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259:
247:
238:
236:
232:
228:
224:
220:
216:
206:
204:
200:
195:
193:
189:
185:
181:
180:star polygons
176:
174:
170:
169:
164:
163:
158:
154:
153:solid polygon
149:
144:
142:
138:
134:
132:
127:
123:
122:
117:
113:
112:
106:
104:
100:
99:line segments
96:
93:
87:
54:
50:
41:
37:
33:
19:
6950:
6919:
6910:
6902:
6893:
6884:
6864:10-orthoplex
6600:Dodecahedron
6521:
6510:
6499:
6490:
6481:
6472:
6468:
6458:
6450:
6446:
6438:
6434:
6168:>20 sides
6103:Decagon (10)
6088:Heptagon (7)
6078:Pentagon (5)
6068:Triangle (3)
5963:Equidiagonal
5873:
5796:
5762:
5756:
5742:
5728:
5720:
5715:
5707:
5702:
5683:
5662:
5652:
5634:
5629:
5611:
5606:
5596:
5591:
5573:
5568:
5550:
5545:
5527:
5522:
5504:
5499:
5489:
5484:
5466:
5461:
5443:
5438:
5417:
5409:
5398:
5397:David Hume,
5393:
5382:
5377:
5358:
5354:
5344:
5332:. Retrieved
5328:Ask Dr. Math
5327:
5302:
5230:
5166:
5161:
5126:
5122:
5116:
5108:
5103:
5071:
5063:
5054:
5048:(18): 23–28.
5045:
5041:
5028:
4917:
4905:
4897:
4892:
4859:math/0408104
4849:
4843:
4834:
4826:
4821:
4807:
4800:
4791:
4785:
4774:the original
4753:
4749:
4736:
4724:. Retrieved
4717:the original
4703:
4694:
4674:
4667:
4657:
4647:
4634:
4627:
4607:
4600:
4591:
4582:Bibliography
4555:Spirolateral
4540:Polygon soup
4428:
4424:
4419:
4413:
4409:
4403:
4399:
4393:
4389:polygon mesh
4386:
4363:
4348:
4342:October 2018
4339:
4328:Please help
4323:verification
4320:
4275:
4265:, or at the
4248:
4241:
4202:
4195:
4183:Aristophanes
4175:star polygon
4164:
4012:
3990:
3935:
3805:triacontagon
3712:heptadecagon
3686:pentadecagon
3673:tetradecagon
3446:
3427:
3424:
3405:
3399:
3395:
3383:
3379:
3371:
3364:
3362:
3351:
3347:
3332:
3270:skew polygon
3240:
3056:
3049:
3042:
3019:
2670:
2654:
2644:
2435:
2431:
2429:
2290:
2286:
2284:
2279:
2273:
2215:
2207:
2205:
2198:
2188:
2184:
2180:
2178:
2171:
2124:
2120:
2118:
2111:
2108:
1608:
1601:
1594:
1583:
1576:
1569:
1565:
1563:
1558:
1552:
1534:
1404:
1196:determinants
1193:
998:
977:
970:
963:
956:
824:
797:
785:
781:
773:
763:
659:
655:
531:
525:
519:
515:
506:
492:
486:
480:
475:
464:
456:
451:
430:
408:star polygon
405:
402:
391:Isotoxal or
380:Isogonal or
365:circumcircle
326:Star polygon
309:
305:
301:
265:
257:
234:
230:
226:
222:
214:
212:
196:
192:skew polygon
177:
172:
167:
166:
161:
160:
156:
152:
145:
143:is a 3-gon.
136:
130:
129:
125:
119:
115:
109:
107:
52:
46:
36:
6873:10-demicube
6834:9-orthoplex
6784:8-orthoplex
6734:7-orthoplex
6691:6-orthoplex
6661:5-orthoplex
6616:Pentachoron
6604:Icosahedron
6579:Tetrahedron
6364:Star-shaped
6339:Rectilinear
6309:Equilateral
6304:Equiangular
6268:Hendecagram
6112:11–20 sides
6093:Octagon (8)
6083:Hexagon (6)
6058:Monogon (1)
5900:Equilateral
4185:, found at
3763:icositrigon
3728:octadecagon
3699:hexadecagon
3559:the plane.
3525:the plane.
3507:the plane.
3497:(or trigon)
3459:Properties
3378:πολύγωνον (
3367:comes from
3333:realization
3320:dimensions.
3255:cartography
1537:orientation
1124:where
711:radians or
582:radians or
421:Rectilinear
345:Equilateral
339:Equiangular
322:dimensions.
292:Star-shaped
97:made up of
6982:Categories
6859:10-simplex
6843:9-demicube
6793:8-demicube
6743:7-demicube
6700:6-demicube
6670:5-demicube
6584:Octahedron
6369:Tangential
6273:Dodecagram
6051:1–10 sides
6042:By number
6023:Tangential
6003:Right kite
4577:References
4271:California
3919:David Hume
3660:tridecagon
3629:hendecagon
3369:Late Latin
3348:polyhedron
3340:polyhedron
1194:or, using
509:− 2) × 180
454:-gon into
435:orthogonal
371:Tangential
182:and other
6907:orthoplex
6829:9-simplex
6779:8-simplex
6729:7-simplex
6686:6-simplex
6656:5-simplex
6625:Tesseract
6349:Reinhardt
6258:Enneagram
6248:Heptagram
6238:Pentagram
6205:65537-gon
6063:Digon (2)
6033:Trapezoid
5998:Rectangle
5948:Bicentric
5910:Isosceles
5887:Triangles
5798:MathWorld
5793:"Polygon"
5361:: 33–59.
5098:Mathworld
5001:⋅
4998:π
4986:π
4977:
4971:⋅
4958:⋅
4943:∞
4937:→
4840:Pak, Igor
4601:Polyhedra
4412:+ 1) / 2(
4382:materials
4370:primitive
4282:honeycomb
4223:In nature
4213:imaginary
4203:In 1952,
4171:pentagram
3978:apeirogon
3964:1,000,000
3945:65537-gon
3902:chiliagon
3647:dodecagon
3540:pentagram
3440:pentagram
3407:dodecagon
3390:-derived
3372:polygōnum
3363:The word
3344:polytopes
3318:imaginary
3281:apeirogon
3203:−
3185:∑
3122:−
3104:∑
3045:triangles
2973:−
2901:−
2883:∑
2801:−
2729:−
2711:∑
2616:−
2609:α
2604:
2598:⋅
2589:−
2582:α
2577:
2571:⋅
2556:−
2549:α
2544:
2508:π
2503:
2446:α
2410:π
2405:
2399:⋅
2391:π
2386:
2380:⋅
2374:⋅
2352:π
2343:
2337:⋅
2324:⋅
2253:⋅
2247:⋅
2154:π
2123:and area
2073:−
2066:θ
2059:
2048:−
2029:−
2015:⋯
1992:−
1985:θ
1978:⋯
1966:θ
1959:
1948:−
1934:⋯
1919:θ
1906:θ
1899:
1871:θ
1864:
1815:−
1808:θ
1801:⋯
1789:θ
1776:θ
1769:
1758:−
1744:⋯
1729:θ
1716:θ
1709:
1681:θ
1674:
1564:The area
1270:−
1252:∑
1243:−
1225:∑
1090:−
1053:−
1035:∑
933:−
914:−
897:…
790:pentagram
732:−
683:−
674:π
593:−
570:π
550:−
461:triangles
213:The word
209:Etymology
18:Polygonal
6988:Polygons
6961:Topics:
6924:demicube
6889:polytope
6883:Uniform
6644:600-cell
6640:120-cell
6593:Demicube
6567:Pentagon
6547:Triangle
6324:Isotoxal
6319:Isogonal
6263:Decagram
6253:Octagram
6243:Hexagram
6044:of sides
5973:Harmonic
5874:Polygons
5691:Archived
5660:(1981).
4655:(1974).
4535:Polyform
4493:See also
4374:vertices
4290:hexagons
4284:made by
4244:crystals
4146:-ennea-
4118:-hepta-
4090:-penta-
4076:-tetra-
3927:myriagon
3751:icosagon
3565:heptagon
3531:pentagon
3495:triangle
3436:pentagon
3412:triangle
3401:pentagon
3352:polytope
3316:and two
3063:centroid
2667:Centroid
1589:and the
796:, where
760:degrees.
654:-gon (a
534:-gon is
489:-gon is
427:Monotone
406:regular
199:polytope
141:triangle
121:vertices
49:geometry
6898:simplex
6868:10-cube
6635:24-cell
6621:16-cell
6562:Hexagon
6416:regular
6344:Regular
6289:Concave
6282:Classes
6190:257-gon
6013:Rhombus
5953:Crossed
5403:p. 101.
5153:1343696
5145:2161556
5082:. p. 4.
4884:6756387
4876:2128993
4770:2686282
4525:Golygon
4278:biology
4153:History
4132:-octa-
4104:-hexa-
4031:-hena-
3960:megagon
3887:257-gon
3617:decagon
3603:nonagon
3591:octagon
3548:hexagon
3465:monogon
3430:regular
3420:nonagon
3398:, e.g.
3365:polygon
3312:of two
3261:of the
2658:figure.
2655:density
2293:of its
2276:apothem
2210:of its
2169:holds.
1607:, ...,
1582:, ...,
802:density
800:is the
512:degrees
501:radians
495:− 2) ×
351:Regular
320:complex
316:Hilbert
302:complex
286:Concave
215:polygon
173:polygon
126:corners
90:) is a
53:polygon
6838:9-cube
6788:8-cube
6738:7-cube
6695:6-cube
6665:5-cube
6552:Square
6429:Family
6354:Simple
6299:Cyclic
6294:Convex
6018:Square
5958:Cyclic
5920:Obtuse
5915:Kepler
5670:
5643:
5620:
5582:
5559:
5536:
5513:
5475:
5452:
5426:
5387:p. 22.
5238:
5151:
5143:
5078:
4882:
4874:
4768:
4682:
4565:Tiling
4489:test.
4255:basalt
4179:krater
4062:-tri-
4025:-kai-
3993:Kepler
3949:65,537
3931:10,000
3773:neusis
3456:Sides
3410:. The
3359:Naming
3061:. The
3059:> 3
2127:, the
1405:where
1001:simple
472:Angles
361:circle
357:Cyclic
306:simple
280:Simple
271:Convex
95:figure
6557:p-gon
6329:Magic
5925:Right
5905:Ideal
5895:Acute
5334:3 May
5141:JSTOR
5038:(PDF)
4880:S2CID
4854:arXiv
4813:(PDF)
4777:(PDF)
4766:JSTOR
4746:(PDF)
4726:6 Feb
4720:(PDF)
4713:(PDF)
4619:Notes
4235:, in
4187:Caere
4048:-di-
4034:-gon
4017:Ones
4009:Tens
3577:with
3478:digon
3453:Name
3388:Greek
3376:Greek
3304:is a
3251:digon
969:) = (
227:gōnía
223:polús
219:Greek
128:. An
116:sides
111:edges
92:plane
6915:cube
6588:Cube
6418:and
6359:Skew
5983:Kite
5878:List
5668:ISBN
5641:ISBN
5618:ISBN
5580:ISBN
5557:ISBN
5534:ISBN
5511:ISBN
5473:ISBN
5450:ISBN
5424:ISBN
5336:2015
5236:ISBN
5076:ISBN
5046:2015
4728:2013
4680:ISBN
4286:bees
4251:lava
4231:The
4209:real
3906:1000
3557:tile
3555:Can
3523:tile
3505:tile
3433:star
3418:and
3396:-gon
3350:and
3314:real
3043:For
2218:by
2148:>
1484:and
1005:area
813:Area
778:turn
528:− 2)
231:gónu
157:body
133:-gon
51:, a
6464:(p)
5767:doi
5363:doi
5306:by
5276:doi
5131:doi
5127:124
4974:sin
4930:lim
4864:doi
4758:doi
4612:pdf
4396:+ 1
4364:In
4332:by
4276:In
4269:in
4261:in
4181:by
4013:and
3891:257
3879:100
3323:An
3279:An
3052:= 3
2601:cos
2574:sin
2541:cot
2500:cot
2402:cos
2383:sin
2340:sin
2056:sin
1956:sin
1896:sin
1861:sin
1766:sin
1706:sin
1671:sin
1557:or
1007:is
804:or
723:180
599:360
590:180
503:or
459:− 2
235:gon
165:or
124:or
114:or
47:In
6984::
6969:•
6965:•
6945:21
6941:•
6938:k1
6934:•
6931:k2
6909:•
6866:•
6836:•
6814:21
6810:•
6807:41
6803:•
6800:42
6786:•
6764:21
6760:•
6757:31
6753:•
6750:32
6736:•
6714:21
6710:•
6707:22
6693:•
6663:•
6642:•
6623:•
6602:•
6586:•
6518:/
6507:/
6497:/
6488:/
6466:/
5795:.
5359:32
5357:.
5353:.
5326:.
5315:^
5286:^
5267:^
5250:^
5174:^
5149:MR
5147:.
5139:.
5125:.
5087:^
5074:,
5044:.
5040:.
4878:.
4872:MR
4870:.
4862:.
4850:34
4848:.
4764:.
4754:17
4752:.
4748:.
4590:;
4384:.
4273:.
4219:.
4193:.
4137:90
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4067:40
4053:30
4039:20
4028:1
3917:,
3913:,
3869:90
3859:80
3849:70
3839:60
3829:50
3819:40
3809:30
3796:25
3786:24
3775:.
3768:23
3755:20
3743:19
3733:18
3717:17
3704:16
3691:15
3678:14
3665:13
3652:12
3634:11
3621:10
3585:.
3489:.
3442:.
3414:,
3404:,
3338:A
3300:A
3293:A
3286:A
3268:A
3245:A
2282:.
2203:.
2181:do
2125:A
1600:,
1593:,
1575:,
1561:.
1208:16
976:,
962:,
411:.
237:.
146:A
105:.
6953:-
6951:n
6943:k
6936:2
6929:1
6922:-
6920:n
6913:-
6911:n
6905:-
6903:n
6896:-
6894:n
6887:-
6885:n
6812:4
6805:2
6798:1
6762:3
6755:2
6748:1
6712:2
6705:1
6534:n
6532:H
6525:2
6522:G
6514:4
6511:F
6503:8
6500:E
6494:7
6491:E
6485:6
6482:E
6473:n
6469:D
6462:2
6459:I
6451:n
6447:B
6439:n
6435:A
6407:e
6400:t
6393:v
5880:)
5876:(
5866:e
5859:t
5852:v
5801:.
5773:.
5769::
5750:.
5736:.
5676:.
5647:.
5624:.
5586:.
5563:.
5540:.
5517:.
5479:.
5456:.
5432:.
5371:.
5365::
5338:.
5278::
5244:.
5155:.
5133::
5023:.
5009:2
5005:R
4995:=
4990:n
4983:2
4966:2
4963:n
4953:2
4949:R
4940:+
4934:n
4886:.
4866::
4856::
4815:.
4760::
4730:.
4688:.
4614:)
4473:)
4468:0
4464:y
4460:,
4455:0
4451:x
4447:(
4444:=
4441:P
4420:n
4416:)
4414:n
4410:n
4408:(
4404:n
4400:n
4394:n
4355:)
4349:(
4344:)
4340:(
4326:.
4143:9
4129:8
4115:7
4101:6
4087:5
4073:4
4059:3
4045:2
3982:∞
3608:9
3595:8
3570:7
3552:6
3535:5
3518:4
3500:3
3482:2
3469:1
3265:.
3221:.
3216:i
3212:y
3206:1
3200:n
3195:0
3192:=
3189:i
3179:n
3176:1
3171:=
3166:y
3162:c
3140:,
3135:i
3131:x
3125:1
3119:n
3114:0
3111:=
3108:i
3098:n
3095:1
3090:=
3085:x
3081:c
3067:n
3057:n
3050:n
3047:(
3028:A
3005:.
3002:)
2997:i
2993:y
2987:1
2984:+
2981:i
2977:x
2968:1
2965:+
2962:i
2958:y
2952:i
2948:x
2944:(
2941:)
2936:1
2933:+
2930:i
2926:y
2922:+
2917:i
2913:y
2909:(
2904:1
2898:n
2893:0
2890:=
2887:i
2876:A
2873:6
2869:1
2864:=
2859:y
2855:C
2833:,
2830:)
2825:i
2821:y
2815:1
2812:+
2809:i
2805:x
2796:1
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2772:(
2769:)
2764:1
2761:+
2758:i
2754:x
2750:+
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2741:x
2737:(
2732:1
2726:n
2721:0
2718:=
2715:i
2704:A
2701:6
2697:1
2692:=
2687:x
2683:C
2625:.
2619:2
2613:n
2592:2
2586:n
2568:n
2565:=
2559:2
2553:n
2536:4
2530:2
2526:s
2522:n
2516:=
2511:n
2495:4
2489:2
2485:s
2481:n
2475:=
2472:A
2449:,
2436:s
2432:n
2413:n
2394:n
2377:n
2369:2
2365:R
2361:=
2356:n
2349:2
2332:2
2329:n
2319:2
2315:R
2311:=
2308:A
2291:R
2287:n
2280:a
2259:.
2256:r
2250:p
2241:2
2238:1
2232:=
2229:A
2216:p
2208:r
2189:n
2185:n
2157:A
2151:4
2143:2
2139:p
2121:p
2090:.
2087:)
2084:]
2081:)
2076:2
2070:n
2062:(
2051:1
2045:n
2041:a
2037:[
2032:2
2026:n
2022:a
2018:+
2012:+
2003:]
2000:)
1995:2
1989:n
1981:+
1975:+
1970:2
1962:(
1951:1
1945:n
1941:a
1937:+
1931:+
1928:)
1923:3
1915:+
1910:2
1902:(
1891:4
1887:a
1883:+
1880:)
1875:2
1867:(
1856:3
1852:a
1848:[
1843:2
1839:a
1835:+
1826:]
1823:)
1818:2
1812:n
1804:+
1798:+
1793:2
1785:+
1780:1
1772:(
1761:1
1755:n
1751:a
1747:+
1741:+
1738:)
1733:2
1725:+
1720:1
1712:(
1701:3
1697:a
1693:+
1690:)
1685:1
1677:(
1666:2
1662:a
1658:[
1653:1
1649:a
1645:(
1640:2
1637:1
1632:=
1629:A
1611:n
1609:θ
1605:2
1602:θ
1598:1
1595:θ
1586:n
1584:a
1580:2
1577:a
1573:1
1570:a
1566:A
1545:y
1541:x
1521:.
1518:)
1513:j
1509:y
1505:,
1500:j
1496:x
1492:(
1472:)
1467:i
1463:y
1459:,
1454:i
1450:x
1446:(
1424:j
1421:,
1418:i
1414:Q
1390:,
1385:|
1377:1
1374:+
1371:j
1368:,
1365:1
1362:+
1359:i
1355:Q
1347:j
1344:,
1341:1
1338:+
1335:i
1331:Q
1321:1
1318:+
1315:j
1312:,
1309:i
1305:Q
1297:j
1294:,
1291:i
1287:Q
1280:|
1273:1
1267:n
1262:0
1259:=
1256:j
1246:1
1240:n
1235:0
1232:=
1229:i
1221:=
1216:2
1212:A
1179:,
1174:0
1170:y
1166:=
1161:n
1157:y
1146:0
1142:x
1138:=
1133:n
1129:x
1119:)
1114:i
1110:y
1104:1
1101:+
1098:i
1094:x
1085:1
1082:+
1079:i
1075:y
1069:i
1065:x
1061:(
1056:1
1050:n
1045:0
1042:=
1039:i
1029:2
1026:1
1021:=
1018:A
983:)
981:0
978:y
974:0
971:x
966:n
964:y
959:n
957:x
955:(
941:)
936:1
930:n
926:y
922:,
917:1
911:n
907:x
903:(
900:,
894:,
891:)
886:1
882:y
878:,
873:1
869:x
865:(
862:,
859:)
854:0
850:y
846:,
841:0
837:x
833:(
798:d
786:d
782:n
774:n
745:p
741:)
738:q
735:2
729:p
726:(
696:p
692:)
689:q
686:2
680:p
677:(
660:q
656:p
639:q
636:p
602:n
566:)
559:n
556:2
547:1
543:(
532:n
526:n
524:(
520:n
516:n
507:n
505:(
497:π
493:n
491:(
487:n
457:n
452:n
431:L
377:.
367:.
137:n
131:n
86:/
83:n
80:ɒ
77:ɡ
74:ɪ
71:l
68:ɒ
65:p
62:ˈ
59:/
55:(
34:.
20:)
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