From Wikipedia, the free encyclopedia
An assortment of polygons
.^ A polygon is a closed figure composed of three or more line segments that intersect at their endpoints.- Beginning Algebra Tutorial on Basic Geometry 12 January 2010 2:53 UTC www.wtamu.edu [Source type: FILTERED WITH BAYES]
^ The symbol , which includes the arrow heads at both ends indicates the whole line where , which does not have the arrow heads, indicates a line segment, which is finite in length (only the part of the line from A to B).- Beginning Algebra Tutorial on Basic Geometry 12 January 2010 2:53 UTC www.wtamu.edu [Source type: FILTERED WITH BAYES]
^ The 'polyline' element defines a set of connected straight line segments.- Basic Shapes - SVG 1.1 - 20030114 12 January 2010 2:53 UTC www.w3.org [Source type: Reference]
.^ Each line segment is called the side .- Beginning Algebra Tutorial on Basic Geometry 12 January 2010 2:53 UTC www.wtamu.edu [Source type: FILTERED WITH BAYES]
^ What could be inside a two-sided polygon?
^ In all cases, the polygon is specified as an array of X,Y coordinates of the corner points.
The interior of the polygon is sometimes called its
body.
.^ For this method, num points sets the minimum number of points that a generated polygon will have (if the polygon were to have fewer, it will not be generated at all).- Priism Help: EditPolygon 11 September 2009 9:28 UTC www.msg.ucsf.edu [Source type: FILTERED WITH BAYES]
.^ Polygon is from the Greek roots poli (many) and gonus (knees) and, interprets literally as many angled.- Origins of some arithmetic terms-2 12 January 2010 2:53 UTC www.pballew.net [Source type: FILTERED WITH BAYES]
^ Question 254550 : how many sides does a polygon have if thhe sum of its interior angles is 540?- Questions on Geometry: Polygons answered by real tutors! 12 January 2010 2:53 UTC www.algebra.com [Source type: FILTERED WITH BAYES]
^ The word still retains that meaning today in phonetics for an unvoiced consonant (as opposed to a voiced consonant, a sonant).- Origins of some arithmetic terms-2 12 January 2010 2:53 UTC www.pballew.net [Source type: FILTERED WITH BAYES]
.^ Alternate interior angles are = *Straight angle = 180 .- Beginning Algebra Tutorial on Basic Geometry 12 January 2010 2:53 UTC www.wtamu.edu [Source type: FILTERED WITH BAYES]
^ The symbol , which includes the arrow heads at both ends indicates the whole line where , which does not have the arrow heads, indicates a line segment, which is finite in length (only the part of the line from A to B).- Beginning Algebra Tutorial on Basic Geometry 12 January 2010 2:53 UTC www.wtamu.edu [Source type: FILTERED WITH BAYES]
^ A straight angle is one that measures exactly 180 degrees: .- Beginning Algebra Tutorial on Basic Geometry 12 January 2010 2:53 UTC www.wtamu.edu [Source type: FILTERED WITH BAYES]
The basic geometrical notion has been adapted in various ways to suit particular purposes.
.^ There are several toggles in the EditPolygon dialog which control how polygon graphics are displayed in the image window.- Priism Help: EditPolygon 11 September 2009 9:28 UTC www.msg.ucsf.edu [Source type: FILTERED WITH BAYES]
^ Because of the nature of the search, it is not necessary for the seed point to lie within the polygon which is generated.- Priism Help: EditPolygon 11 September 2009 9:28 UTC www.msg.ucsf.edu [Source type: FILTERED WITH BAYES]
^ Auto Clicking in the image window causes the point selected to be used as the starting point for automatically generating a polygon.- Priism Help: EditPolygon 11 September 2009 9:28 UTC www.msg.ucsf.edu [Source type: FILTERED WITH BAYES]
Classification
Number of sides
.^ The applet shows a regular polygon where the user can drag the vertices to reshape it and alter the number of sides.- Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]
^ The applet shows a polygon where the user can drag any vertex and change the number of sides.- Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]
^ The user can also alter the number of sides from 3 to 99, the title changing to reflect it's name up to 12 sides.- Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]
Convexity
Polygons may be characterised by their degree of convexity:
.^ The polygon that is generated can be larger or smaller than the boundary found using any of the methods.- Priism Help: EditPolygon 11 September 2009 9:28 UTC www.msg.ucsf.edu [Source type: FILTERED WITH BAYES]
^ Rod may chime in here, he knows much more detail about the program than I do.- Autodesk: Discussion Groups - WHY CAN'T I DRAFT WITH AUTOSKETCH 9? 12 January 2010 2:53 UTC discussion.autodesk.com [Source type: FILTERED WITH BAYES]
^ If this transform represents a non-uniform scale or more general transform then the determinant is not likely to represent a value useful for any purpose other than determining if inverse transforms are possible.- AffineTransform (Java 2 Platform SE v1.4.2) 11 September 2009 9:28 UTC java.sun.com [Source type: Reference]
.^ When on, all polygons in the currently displayed z section and time point are displayed, regardless of the wave.- Priism Help: EditPolygon 11 September 2009 9:28 UTC www.msg.ucsf.edu [Source type: FILTERED WITH BAYES]
^ Smoothes the currently selected polygon by operating on all the jagged edges at once.- Priism Help: EditPolygon 11 September 2009 9:28 UTC www.msg.ucsf.edu [Source type: FILTERED WITH BAYES]
^ When EditPolygon is in the autopolygon mode, rather than manually selecting all the vertices of a polygon, a seed point is selected and that is used to generate the polygon (or polygons if a 3D search is enabled).- Priism Help: EditPolygon 11 September 2009 9:28 UTC www.msg.ucsf.edu [Source type: FILTERED WITH BAYES]
The polygon must be simple, and may be convex or concave.
Self-intersecting: the boundary of the polygon crosses itself. Branko Grünbaum calls these coptic, though this term does not seem to be widely used. The term complex is sometimes used in contrast to simple, but this risks confusion with the idea of a complex polygon as one which exists in the complex Hilbert plane consisting of two complex dimensions.
Star polygon: a polygon which self-intersects in a regular way.
Symmetry
- Equiangular: all its corner angles are equal.
- Cyclic: all corners lie on a single circle.
- Isogonal or vertex-transitive: all corners lie within the same symmetry orbit. .
- Equilateral: all edges are of the same length.^ Smoothes the currently selected polygon by operating on all the jagged edges at once.
- Priism Help: EditPolygon 11 September 2009 9:28 UTC www.msg.ucsf.edu [Source type: FILTERED WITH BAYES]
^ It uses the same criteria as change from center to locate a point on the polygon edge to the right of the seed point.- Priism Help: EditPolygon 11 September 2009 9:28 UTC www.msg.ucsf.edu [Source type: FILTERED WITH BAYES]
^ All of these programs can look at the same set of polygons so that they can be used together.- Priism Help: EditPolygon 11 September 2009 9:28 UTC www.msg.ucsf.edu [Source type: FILTERED WITH BAYES]
(A polygon with 5 or more sides can be equilateral without being convex.) (Williams 1979, pp. .
- Isotoxal or edge-transitive: all sides lie within the same symmetry orbit.^ This method computes a binary OR of the appropriate mask values indicating, for each side of this Rectangle , whether or not the specified coordinates are on the same side of the edge as the rest of this Rectangle .
- Rectangle (Java 2 Platform SE v1.4.2) 11 September 2009 9:28 UTC java.sun.com [Source type: FILTERED WITH BAYES]
The polygon is also equilateral.
- Regular. A polygon is regular if it is both cyclic and equilateral. A non-convex regular polygon is called a regular star polygon.
Miscellaneous
- Rectilinear: a polygon whose sides meet at right angles, i.e., all its interior angles are 90 or 270 degrees.
- Monotone with respect to a given line L, if every line orthogonal to L intersects the polygon not more than twice.
Properties
Angles
.^ The applet shows a regular polygon where the user can drag the vertices to reshape it and alter the number of sides.- Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]
^ The applet shows a regular polygon where the user can alter the number of sides and resize it by dragging any vertex.- Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]
^ The user can drag any vertex, change the number of sides in the range 3..99, and make it regular or irregular.- Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]
Each corner has several angles. The two most important ones are:
.^ The applet shows an irregular polygon initially with one interior angle greater than 180 degrees.- Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]
^ The web page lists the properties of a dodecagon including interior angles, exterior angles, sum of exterior angles, area, number of diagonals and number of internal triangles.- Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]
^ (Complete Item Description) Abstract: An interactive applet and associated web page that demonstrate the concept of a convex polygon - one where all interior angle are less than 180 degrees.- Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]
.^ The applet shows an irregular polygon initially with one interior angle greater than 180 degrees.- Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]
^ The web page lists the properties of a dodecagon including interior angles, exterior angles, sum of exterior angles, area, number of diagonals and number of internal triangles.- Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]
^ The web page lists the properties of an octagon including interior angles, exterior angles, sum of exterior angles, area, number of diagonals and number of internal triangles.- Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]
.^ The applet shows an irregular polygon initially with one interior angle greater than 180 degrees.- Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]
^ User can see that the interior and exterior angles are constant in a regular octagon, but vary in an irregular version.- Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]
^ User can see that the interior and exterior angles are constant in a regular pentagon, but vary in an irregular version.- Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]
.^ Interior / exterior angles of a polygon .- Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]
^ The applet shows an irregular polygon initially with one interior angle greater than 180 degrees.- Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]
^ The web page lists the properties of a dodecagon including interior angles, exterior angles, sum of exterior angles, area, number of diagonals and number of internal triangles.- Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]
The amount you "turn" at a corner is the exterior or external angle.
.^ Interior / exterior angles of a polygon .- Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]
^ The applet shows an irregular polygon initially with one interior angle greater than 180 degrees.- Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]
^ That one is great fun and familiarity with it makes you a pro.- Autodesk: Discussion Groups - WHY CAN'T I DRAFT WITH AUTOSKETCH 9? 12 January 2010 2:53 UTC discussion.autodesk.com [Source type: FILTERED WITH BAYES]
.^ At least I can make a new one, move an entity to it, turn it on and off.- Autodesk: Discussion Groups - WHY CAN'T I DRAFT WITH AUTOSKETCH 9? 12 January 2010 2:53 UTC discussion.autodesk.com [Source type: FILTERED WITH BAYES]
720° for a
pentagram and 0° for an angular "eight", where
d is the density or starriness of the polygon. See also
orbit (dynamics).
.^ Interior / exterior angles of a polygon .- Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]
^ The web page lists the properties of a dodecagon including interior angles, exterior angles, sum of exterior angles, area, number of diagonals and number of internal triangles.- Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]
^ User can see that the interior and exterior angles are constant in a regular octagon, but vary in an irregular version.- Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]
.^ An object thus beaten becomes blunt, dull, or rounded, as in the application to an obtuse angle, one having more than 90 o but less than 180 o .- Origins of some arithmetic terms-2 12 January 2010 2:53 UTC www.pballew.net [Source type: FILTERED WITH BAYES]
^ The applet shows an irregular polygon initially with one interior angle greater than 180 degrees.- Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]
^ Some feared John Kerry, others John Edwards, because his personality wears well over time, and others even Bob Graham, because he can carry Florida, more than Dean.- Polygon, the Dancing Bear 12 January 2010 2:53 UTC potifos.com [Source type: General]
.^ The applet shows a polygon where the user can drag any vertex and change the number of sides.- Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]
^ The user can drag any vertex and change the number of sides in the range 3..99.- Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]
^ The applet shows a regular polygon where the user can alter the number of sides and resize it by dragging any vertex.- Browse: Keywords: Polygon | OER Commons 12 January 2010 2:53 UTC www.oercommons.org [Source type: General]
)
Area and centroid
Nomenclature of a 2D polygon.
.^ Commands BOUNDARY Creates a region or a polyline from an enclosed area.- Autodesk: Discussion Groups - WHY CAN'T I DRAFT WITH AUTOSKETCH 9? 12 January 2010 2:53 UTC discussion.autodesk.com [Source type: FILTERED WITH BAYES]
For a non-self-intersecting (
simple) polygon with
n vertices, the area and
centroid are given by
[1]:



To close the polygon, the first and last vertices are the same, i.e.,
xn,yn = x0,y0. The vertices must be ordered clockwise or counterclockwise; if they are ordered clockwise, the area will be negative but correct in
absolute value. This is commonly called the Surveyor's Formula.
[citation needed]
.^ Second When the pars minutia [see Minute of an arc needed to be divided into even smaller parts, the 1/60 part of 1/60 of a degree needed a name also.- Origins of some arithmetic terms-2 12 January 2010 2:53 UTC www.pballew.net [Source type: FILTERED WITH BAYES]
![\begin{align}A = \frac12 ( a_1[a_2 \sin( heta_1) + a_3 \sin( heta_1 + heta_2) + \cdots + a_{n-1} \sin( heta_1 + heta_2 + \cdots + heta_{n-2})] \ {} + a_2[a_3 \sin( heta_2) + a_4 \sin( heta_2 + heta_3) + \cdots + a_{n-1} \sin( heta_2 + \cdots + heta_{n-2})] \ {} + \cdots + a_{n-2}[a_{n-1} \sin( heta_{n-2})] ) \end{align}](http://images-mediawiki-sites.thefullwiki.org/11/4/2/5/54121954117144167.png)
The formula was described by Lopshits in 1963.
[2]
.^ When on, all polygons in the currently displayed z section and time point are displayed, regardless of the wave.- Priism Help: EditPolygon 11 September 2009 9:28 UTC www.msg.ucsf.edu [Source type: FILTERED WITH BAYES]
^ When EditPolygon is in the autopolygon mode, rather than manually selecting all the vertices of a polygon, a seed point is selected and that is used to generate the polygon (or polygons if a 3D search is enabled).- Priism Help: EditPolygon 11 September 2009 9:28 UTC www.msg.ucsf.edu [Source type: FILTERED WITH BAYES]
^ The determination of the seed point from the existing polygon is controlled by the using last picked pt toggle.- Priism Help: EditPolygon 11 September 2009 9:28 UTC www.msg.ucsf.edu [Source type: FILTERED WITH BAYES]
.^ These polygons are used to delineate areas of the data for measurements, for cutting out part of the image, or as building blocks for 3D Objects which can be used for 3D measurement and modeling.- Priism Help: EditPolygon 11 September 2009 9:28 UTC www.msg.ucsf.edu [Source type: FILTERED WITH BAYES]
^ If you have two overlapping circles and you drop a color in the overlapping area, only that overlapping area is filled, makes it easy to fill overlapping polygons with contrasting colors.- Autodesk: Discussion Groups - WHY CAN'T I DRAFT WITH AUTOSKETCH 9? 12 January 2010 2:53 UTC discussion.autodesk.com [Source type: FILTERED WITH BAYES]
^ It traces a polygon with a hidden line around the perimeter of the blank area you drop the fill into.- Autodesk: Discussion Groups - WHY CAN'T I DRAFT WITH AUTOSKETCH 9? 12 January 2010 2:53 UTC discussion.autodesk.com [Source type: FILTERED WITH BAYES]
This is the
Bolyai-Gerwien theorem.
For a regular polygon with n sides of length s, the area is given by:

Self-intersecting polygons
.^ I had no trouble locating the following description of hatching: " You can hatch an enclosed area or hatch within a specified boundary using HATCH. By default, HATCH creates associative hatches that are updated when the boundary is changed.- Autodesk: Discussion Groups - WHY CAN'T I DRAFT WITH AUTOSKETCH 9? 12 January 2010 2:53 UTC discussion.autodesk.com [Source type: FILTERED WITH BAYES]
^ Use the method option menu in the autopolygon dialog to select the type of algorithm used to generate a polygon from a seed point.- Priism Help: EditPolygon 11 September 2009 9:28 UTC www.msg.ucsf.edu [Source type: FILTERED WITH BAYES]
^ This is equivalent to calling concatenate(SH), where SH is an AffineTransform represented by the following matrix: [ 1 shx 0 ] [ shy 1 0 ] [ 0 0 1 ] Parameters: shx - the multiplier by which coordinates are shifted in the direction of the positive X axis as a factor of their Y coordinate shy - the multiplier by which coordinates are shifted in the direction of the positive Y axis as a factor of their X coordinate .- AffineTransform (Java 2 Platform SE v1.4.2) 11 September 2009 9:28 UTC java.sun.com [Source type: Reference]
.^ In each of these two instances the lines defining the areas to be filled are a mix of whole line entities and portions of larger line entities.- Autodesk: Discussion Groups - WHY CAN'T I DRAFT WITH AUTOSKETCH 9? 12 January 2010 2:53 UTC discussion.autodesk.com [Source type: FILTERED WITH BAYES]
^ Invalid Hatch Boundaries When a hatch boundary cannot be determined, it might be because the specified internal point is not within a fully enclosed area.- Autodesk: Discussion Groups - WHY CAN'T I DRAFT WITH AUTOSKETCH 9? 12 January 2010 2:53 UTC discussion.autodesk.com [Source type: FILTERED WITH BAYES]
^ Another point: improvement in transportation and communications have increasingly centralized our news and our culture at the national level, while eroding local and regional culture or even awareness.- Polygon, the Dancing Bear 12 January 2010 2:53 UTC potifos.com [Source type: General]
This corresponds to the area of the plane covered by the polygon, or to the area of a simple polygon having the same outline as the self-intersecting one (or, in the case of the cross-quadrilateral, the two simple triangles).
Degrees of freedom
An
n-gon has 2
n degrees of freedom, including 2 for position, 1 for rotational orientation, and 1 for over-all size, so 2
n − 4 for
shape. In the case of a
line of symmetry the latter reduces to
n − 2.
.^ There are two major types of self symmetry, rotational (point) and reflective (line).- Origins of some arithmetic terms-2 12 January 2010 2:53 UTC www.pballew.net [Source type: FILTERED WITH BAYES]
With additional mirror-image symmetry (
Dk) there are
n − 1 degrees of freedom.
Generalizations of polygons
In a broad sense, a polygon is an unbounded (without ends) sequence or circuit of alternating segments (sides) and angles (corners).
.^ The first exercise, where you draw a 2" by 3" rectangle, goes right to my "putting the pencil back on the paper" question.- Autodesk: Discussion Groups - WHY CAN'T I DRAFT WITH AUTOSKETCH 9? 12 January 2010 2:53 UTC discussion.autodesk.com [Source type: FILTERED WITH BAYES]
.^ Today in mathematics we define a sequence as a group of terms in a row such as 2,4,6,8.- Origins of some arithmetic terms-2 12 January 2010 2:53 UTC www.pballew.net [Source type: FILTERED WITH BAYES]
^ For example, using the sequence of even terms above, the sequence of partial sums would be 2, 2+4, 2+4+6, etc.- Origins of some arithmetic terms-2 12 January 2010 2:53 UTC www.pballew.net [Source type: FILTERED WITH BAYES]
The interior (body) of the polygon is another element, and (for technical reasons) so is the null polytope or nullitope.
A geometric polygon is understood to be a "realization" of the associated abstract polygon; this involves some "mapping" of elements from the abstract to the geometric.
.^ If you have two overlapping circles and you drop a color in the overlapping area, only that overlapping area is filled, makes it easy to fill overlapping polygons with contrasting colors.- Autodesk: Discussion Groups - WHY CAN'T I DRAFT WITH AUTOSKETCH 9? 12 January 2010 2:53 UTC discussion.autodesk.com [Source type: FILTERED WITH BAYES]
For example a
spherical polygon is drawn on the surface of a sphere, and its sides are arcs of great circles.
.^ We are only talking about resolution here, bigger characters do not necessarily require more polygons.- Nintendo May Be Considering SD/HD Hybrid Console 12 January 2010 2:53 UTC www.escapistmagazine.com [Source type: General]
A
digon is a closed polygon having two sides and two corners. On the sphere, we can mark two opposing points (like the North and South poles) and join them by half a great circle.
.^ Polygons inside polygons To put different fills in concentric polygons (you would have to convert the circles in a donut) select the inside polygon Edit> Trim> Difference and select the outside polygon.- Autodesk: Discussion Groups - WHY CAN'T I DRAFT WITH AUTOSKETCH 9? 12 January 2010 2:53 UTC discussion.autodesk.com [Source type: FILTERED WITH BAYES]
Tile the sphere with digons and you have a
polyhedron called a
hosohedron.
.^ If the third toggle on the left (the one next to the wave field) is on, the copy is done from all time points z sections in the wave given by the wave field on the left.- Priism Help: EditPolygon 11 September 2009 9:28 UTC www.msg.ucsf.edu [Source type: FILTERED WITH BAYES]
^ That one is great fun and familiarity with it makes you a pro.- Autodesk: Discussion Groups - WHY CAN'T I DRAFT WITH AUTOSKETCH 9? 12 January 2010 2:53 UTC discussion.autodesk.com [Source type: FILTERED WITH BAYES]
^ If it is set to one , then the polygons you create will only be applied to the currently active wave.- Priism Help: EditPolygon 11 September 2009 9:28 UTC www.msg.ucsf.edu [Source type: FILTERED WITH BAYES]
.^ When any of one of these programs are running and have a set of polygons loaded into an image window, any other one will see those polygons and be able to work with them.- Priism Help: EditPolygon 11 September 2009 9:28 UTC www.msg.ucsf.edu [Source type: FILTERED WITH BAYES]
The idea of a polygon has been generalized in various ways. Here is a short list of some
degenerate cases (or special cases, depending on your point of view):
- Digon. Interior angle of 0° in the Euclidean plane. See remarks above re. on the sphere.
.^ An object thus beaten becomes blunt, dull, or rounded, as in the application to an obtuse angle, one having more than 90 o but less than 180 o .- Origins of some arithmetic terms-2 12 January 2010 2:53 UTC www.pballew.net [Source type: FILTERED WITH BAYES]
The Petrie polygons of the regular polyhedra are classic examples.
- A spherical polygon is a circuit of sides and corners on the surface of a sphere.
- An apeirogon is an infinite sequence of sides and angles, which is not closed but it has no ends because it extends infinitely.
- A complex polygon is a figure analogous to an ordinary polygon, which exists in the complex Hilbert plane.
Naming polygons
.^ The word is a hybrid of Greek and Latin roots.- Origins of some arithmetic terms-2 12 January 2010 2:53 UTC www.pballew.net [Source type: FILTERED WITH BAYES]
^ The word comes from the Latin rota , for wheel.- Origins of some arithmetic terms-2 12 January 2010 2:53 UTC www.pballew.net [Source type: FILTERED WITH BAYES]
^ Prime is from the Latin word for first, primus and related to the Greek protos .- Origins of some arithmetic terms-2 12 January 2010 2:53 UTC www.pballew.net [Source type: FILTERED WITH BAYES]
pentagon,
dodecagon. The
triangle,
quadrilateral or quadrangle, and
nonagon are exceptions. For large numbers,
mathematicians usually write the
numeral itself, e.g.
17-gon. A variable can even be used, usually
n-gon. This is useful if the number of sides is used in a
formula.
Polygon names
Name |
Edges |
Remarks |
henagon (or monogon) |
1 |
In the Euclidean plane, degenerates to a closed curve with a single vertex point on it. |
digon |
2 |
In the Euclidean plane, degenerates to a closed curve with two vertex points on it. |
triangle (or trigon) |
3 |
The simplest polygon which can exist in the Euclidean plane. |
quadrilateral (or quadrangle or tetragon) |
4 |
The simplest polygon which can cross itself. |
pentagon |
5 |
The simplest polygon which can exist as a regular star. A star pentagon is known as a pentagram or pentacle. |
hexagon |
6 |
|
heptagon |
7 |
avoid "septagon" = Latin [sept-] + Greek |
octagon |
8 |
|
enneagon (or nonagon) |
9 |
|
decagon |
10 |
|
hendecagon |
11 |
avoid "undecagon" = Latin [un-] + Greek |
dodecagon |
12 |
avoid "duodecagon" = Latin [duo-] + Greek |
tridecagon (or triskaidecagon) |
13 |
|
tetradecagon (or tetrakaidecagon) |
14 |
|
pentadecagon (or quindecagon or pentakaidecagon) |
15 |
|
hexadecagon (or hexakaidecagon) |
16 |
|
heptadecagon (or heptakaidecagon) |
17 |
|
octadecagon (or octakaidecagon) |
18 |
|
enneadecagon (or enneakaidecagon or nonadecagon) |
19 |
|
icosagon |
20 |
|
No established English name |
100 |
"hectogon" is the Greek name (see hectometre), .^ The word is a hybrid of Greek and Latin roots.- Origins of some arithmetic terms-2 12 January 2010 2:53 UTC www.pballew.net [Source type: FILTERED WITH BAYES]
^ Etymologically, it means "handed", but is a hybrid, since it combines a Greek stem with the Latin suffix "-al".- Origins of some arithmetic terms-2 12 January 2010 2:53 UTC www.pballew.net [Source type: FILTERED WITH BAYES]
|
chiliagon |
1000 |
Pronounced /ˈkɪliəɡɒn/), this polygon has 1000 sides. The measure of each angle in a regular chiliagon is 179.64°.
René Descartes used the chiliagon and myriagon (see below) as examples in his Sixth meditation to demonstrate a distinction which he made between pure intellection and imagination. He cannot imagine all thousand sides [of the chiliagon], as he can for a triangle. However, he clearly understands what a chiliagon is, just as he understands what a triangle is, and he is able to distinguish it from a myriagon. Thus, he claims, the intellect is not dependent on imagination. [3]
|
myriagon |
10,000 |
See remarks on the chiliagon. |
megagon [4] |
1,000,000 |
The internal angle of a regular megagon is 179.99964 degrees. |
To construct the name of a polygon with more than 20 and less than 100 edges, combine the prefixes as follows
Tens |
and |
Ones |
final suffix |
-kai- |
1 |
-hena- |
-gon |
20 |
icosi- |
2 |
-di- |
30 |
triaconta- |
3 |
-tri- |
40 |
tetraconta- |
4 |
-tetra- |
50 |
pentaconta- |
5 |
-penta- |
60 |
hexaconta- |
6 |
-hexa- |
70 |
heptaconta- |
7 |
-hepta- |
80 |
octaconta- |
8 |
-octa- |
90 |
enneaconta- |
9 |
-ennea- |
The "kai" is not always used. Opinions differ on exactly when it should, or need not, be used (see also examples above).
Alternatively, the system used for naming the
higher alkanes can be used:
Ones |
Tens |
final suffix |
1 |
hen- |
10 |
deca- |
-gon |
2 |
do- |
20 |
-cosa- |
3 |
tri- |
30 |
triaconta- |
4 |
tetra- |
40 |
tetraconta- |
5 |
penta- |
50 |
pentaconta- |
6 |
hexa- |
60 |
hexaconta- |
7 |
hepta- |
70 |
heptaconta- |
8 |
octa- |
80 |
octaconta- |
9 |
ennea- (or nona-) |
90 |
enneaconta- (or nonaconta-) |
This has the advantage of being consistent with the system used for 10- thru 19-sided figures.
That is, a 42-sided figure would be named as follows:
Ones |
Tens |
final suffix |
full polygon name |
do- |
tetraconta- |
-gon |
dotetracontagon |
and a 50-sided figure
Tens |
and |
Ones |
final suffix |
full polygon name |
pentaconta- |
|
-gon |
pentacontagon |
But beyond enneagons and decagons, professional mathematicians generally prefer the aforementioned numeral notation (for example,
MathWorld has articles on 17-gons and 257-gons). Exceptions exist for side numbers that are difficult to express in numerical form.
History
Polygons have been known since ancient times. The
regular polygons were known to the ancient Greeks, and the
pentagram, a non-convex regular polygon (
star polygon), appears on the vase of Aristophonus, Caere, dated to the 7th century B.C..
[citation needed] Non-convex polygons in general were not systematically studied until the 14th century by Thomas Bredwardine.
[citation needed]
.^ Mobius work with barycenters present the idea of directed distances, and Argand presentated complex numbers as points or ordered pairs of real numbers.- Origins of some arithmetic terms-2 12 January 2010 2:53 UTC www.pballew.net [Source type: FILTERED WITH BAYES]
Polygons in nature
Numerous regular polygons may be seen in nature. In the world of
geology, crystals have flat faces, or facets, which are polygons.
Quasicrystals can even have regular pentagons as faces.
.^ They go on to state that "[b]y the definition of trapezoid here given it will be seen that the parallelogram may be considered a special form of the trapezoid.- Origins of some arithmetic terms-2 12 January 2010 2:53 UTC www.pballew.net [Source type: FILTERED WITH BAYES]
The most famous hexagons in nature are found in the animal kingdom. The wax
honeycomb made by
bees is an array of
hexagons used to store honey and pollen, and as a secure place for the larvae to grow. There also exist animals who themselves take the approximate form of regular polygons, or at least have the same symmetry. For example,
sea stars display the symmetry of a
pentagon or, less frequently, the
heptagon or other polygons.
.^ Reflective symmetry is sometimes called mirror symmetry because one part of the object looks like the reflection of the other half.- Origins of some arithmetic terms-2 12 January 2010 2:53 UTC www.pballew.net [Source type: FILTERED WITH BAYES]
Though echinoderms do not exhibit exact
radial symmetry,
jellyfish and
comb jellies do, usually fourfold or eightfold.
Radial symmetry (and other symmetry) is also widely observed in the plant kingdom, particularly amongst flowers, and (to a lesser extent) seeds and fruit, the most common form of such symmetry being pentagonal. A particularly striking example is the
Starfruit, a slightly tangy fruit popular in Southeast Asia, whose cross-section is shaped like a pentagonal star.
.^ Two chiral objects which are reflected images of each other are called enantiomorphic , from the Greek words for opposite, anti'os , and body, morph .- Origins of some arithmetic terms-2 12 January 2010 2:53 UTC www.pballew.net [Source type: FILTERED WITH BAYES]
^ She would then cut the monologue short, because there was no point in being excited about your hobbies with such a heavy heart.- Writing.Com: Different and Cool- Love's a Polygon 12 January 2010 2:53 UTC www.writing.com [Source type: Original source]
^ Loculus seems to be a word related to the division of a tomb area into small chambers for different bodies and is related to the diminutive of locus for a point or place, thus "a little place".- Origins of some arithmetic terms-2 12 January 2010 2:53 UTC www.pballew.net [Source type: FILTERED WITH BAYES]
The sun-earth system has five Lagrangian points.
.^ He wasn't exactly two steps ahead, but with every blow, Tarah could see him get more and more comfortable at being swung at; his cool expression testified to that.- Writing.Com: Different and Cool- Love's a Polygon 12 January 2010 2:53 UTC www.writing.com [Source type: Original source]
^ These are the two tools you'll use the most at the beginning.- Autodesk: Discussion Groups - WHY CAN'T I DRAFT WITH AUTOSKETCH 9? 12 January 2010 2:53 UTC discussion.autodesk.com [Source type: FILTERED WITH BAYES]
Astronomers have already found
asteroids at these points.
.^ She would then cut the monologue short, because there was no point in being excited about your hobbies with such a heavy heart.- Writing.Com: Different and Cool- Love's a Polygon 12 January 2010 2:53 UTC www.writing.com [Source type: Original source]
There are already satellites and space observatories at the less stable Lagrangian points.
Uses
- Cut up a piece of paper into polygons, and put them back together as a tangram.
- Join many edge-to-edge as a tiling or tessellation.
- Join several edge-to-edge and fold them all up so there are no gaps, to make a three-dimensional polyhedron.
- Use computer-generated polygons to build up a three-dimensional world full of monsters, theme parks, aeroplanes or anything; see Polygons in computer graphics below.
In computer graphics
A polygon in a
computer graphics (image generation) system is a two-dimensional shape that is modelled and stored within its database. A polygon can be coloured, shaded and textured, and its position in the database is defined by the co-ordinates of its vertices (corners).
Naming conventions differ from those of mathematicians:
- A simple polygon does not cross itself.
- a concave polygon is a simple polygon having at least one interior angle greater than 180°.
- A complex polygon does cross itself.
Use of Polygons in Real-time imagery. 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 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 so that as the viewing point moves through the scene, it is perceived in 3D.
Morphing.
.^ The answer, c, is most often called the difference or result, but in many applied statistical uses it is also called the residue , or residual , that which remains.- Origins of some arithmetic terms-2 12 January 2010 2:53 UTC www.pballew.net [Source type: FILTERED WITH BAYES]
.^ She wondered if it was less that they were trying to win and more that they simply didn't like her.- Writing.Com: Different and Cool- Love's a Polygon 12 January 2010 2:53 UTC www.writing.com [Source type: Original source]
^ The chill that was less like glacier water and more like icy fire, flickering as a blazing knife deep in his heart.- Writing.Com: Different and Cool- Love's a Polygon 12 January 2010 2:53 UTC www.writing.com [Source type: Original source]
Meshed Polygons. The number of meshed polygons ("meshed" is like a fish net) can be up to twice that of free-standing unmeshed polygons, particularly if the polygons are contiguous.
.^ There are two major types of self symmetry, rotational (point) and reflective (line).- Origins of some arithmetic terms-2 12 January 2010 2:53 UTC www.pballew.net [Source type: FILTERED WITH BAYES]
There are (n+1) 2/2n2 vertices per triangle. Where n is large, this approaches one half. Or, each vertex inside the square mesh connects four edges (lines).
Polygon Count.
.^ Then, if I occupy one of the markers, backsight on another, as in the paragraph above, and lay out a bunch of surveyed points, it seems that I will have to rotate the entire grid around the occupied point in order to make the backsight zero degrees.- Autodesk: Discussion Groups - WHY CAN'T I DRAFT WITH AUTOSKETCH 9? 12 January 2010 2:53 UTC discussion.autodesk.com [Source type: FILTERED WITH BAYES]
^ Some geometry textbooks define a trapezoid as a quadrilateral with at least one pair of parallel sides, so that a parallelogram is a type of trapezoid.- Origins of some arithmetic terms-2 12 January 2010 2:53 UTC www.pballew.net [Source type: FILTERED WITH BAYES]
When analysing the characteristics of a particular imaging system, the exact definition of polygon count should be obtained as it applies to that system as there is some flexibility in processing which causes comparisons to become non-trivial.
Vertex Count. Although using this metric appears to be closer to reality it still must be taken with some salt. Since each vertex can be augmented with other attributes (such as color or normal) the amount of processing involved cannot be trivially inferred. Furthermore, the applied vertex transform is to be accounted, as well topology information specific to the system being evaluated as post-transform caching can introduce consistent variations in the expected results.
In popular culture
- They Might Be Giants have a song entitled "Nonagon" on their children's album "Here Come the 123s." The song anthropomorphizes each of the regular polygons with three through eight sides (except the heptagon), placing them at a party hosted by the Nonagon. A video on the DVD featuring this song shows each of the polygons as equiangular shapes with simply-drawn human characteristics.[5]
See also
References
Notes
Bibliography
- Coxeter, H.S.M.; Regular Polytopes, (Methuen and Co., 1948).
- Cromwell, P.;Polyhedra, CUP hbk (1997), pbk. (1999).
- Grünbaum, B.; Are your polyhedra the same as my polyhedra? Discrete and comput. geom: the Goodman-Pollack festschrift, ed. Aronov et al. Springer (2003) pp. 461–488. (pdf)
External links
- Polygon name generator, enter the number of sides to see the polygon's name
- Weisstein, Eric W., "Polygon" from MathWorld.
- What Are Polyhedra?, with Greek Numerical Prefixes
- Polygons, types of polygons, and polygon properties, with interactive animation
- How to draw monochrome orthogonal polygons on screens, by Herbert Glarner
- comp.graphics.algorithms Frequently Asked Questions, solutions to mathematical problems computing 2D and 3D polygons
- Comparison of the different algorithms for Polygon Boolean operations, compares capabilities, speed and numerical robustness
Polygons |
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Listed by number of sides |
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1–10 sides |
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11–20 sides |
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Others |
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Star polygons |
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