File:Construction of an almost regular hexagon and dodecagon..svg

From Wikimedia Commons, the free media repository
Jump to navigation Jump to search

Original file ‎(SVG file, nominally 400 × 400 pixels, file size: 59 KB)

Captions

Captions

Add a one-line explanation of what this file represents
Description Square of the circle (approximation) -- almost regular hexagon constructed by placing vertices on integer points of a 17x17 grid. Based on the fact that arctan(4/7)=29.74488°, a close approximation to 30°. Some of the vertices of the slightly irregular hexagon are 8 units away from the origin, while others are sqrt(4²+7²) or 8.06226...
Date
Source made by me (Inkscape)
Author user:Mjchael
Permission
(Reusing this file)
I, the copyright holder of this work, hereby publish it under the following license:
w:en:Creative Commons
attribution share alike
This file is licensed under the Creative Commons Attribution-Share Alike 2.5 Generic license.
You are free:
  • to share – to copy, distribute and transmit the work
  • to remix – to adapt the work
Under the following conditions:
  • attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
  • share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.
SVG development
InfoField
 
The SVG code is valid.
 
This trigonometry was created with CorelDRAW.
 
The file size of this SVG trigonometry may be irrationally large because its text has been converted to paths inhibiting translations.
llink=Category:Qs icons SVGThis SVG trigonometry shows a very simple image. Drawing uncomplicated graphics with a text editor seems more adequate than using a vector graphics program, and will often result in a dramatic reduction of file size.
Deutsch: Der Name der Grafik besagt, dass es möglich ist, nur durch das Abzählen von quadratischen Rechenkästchen 12 Punkte so auf einer Kreisbahn zu positionieren, dass sich daraus ein annähernd regelmäßiges Zwölfeck ergibt.
Annäherung an 30° durch einen Tangens von 7/4.
Abweichung vom Ideal < 0,8%.

Für viele praktische Zwecke erreicht man eine ausreichende Annäherung an ein reguläres Sechseck, wenn man dieses aus annähernd gleichseitigen Dreiecken konstruiert. Ein annähernd gleichseitiges Dreieck erhält man, wenn die Grundseite des Dreiecks zur Höhe in einem Verhältnis von 8:7 steht. Die Berechnung in der Grafik zeigt, dass die beiden Schenkel des so konstruierten Dreiecks annähernd die Länge der Grundseite besitzen. Diese Annäherung findet dort Anwendung, wo ein Fehler von knapp 1 % toleriert werden kann. So können zum Beispiel Karten oder Spielfelder in leicht zu zeichnende und leicht zu berechnende Sektoren eingeteilt werden. Die Konstruktion gelingt auch problemlos Kindern im Grundschulalter, da nicht mehr benötigt wird, als das Abzählen von Rechenkästchen. Aufwendige Konstruktionen mit Zirkel und Geodreieck entfallen.

Praktischer Nutzen: einfache Graphiken und Design, Kartographie, Sechseckkonstruktionen am Computer bei Zeichenprogrammen ohne Winkelfunktion (ausschließlich mit Punkten).

Für größere Konstruktionen könnte auch ein Verhältnis von 7:12 interessant sein: 72 + 122 = 193 = 13,89...2 ~ 196 = 142 (Abweichung < 0,8%)

Vektorkoordinaten

Die Punkte Pi (x|y) sind im folgenden nach den Ziffern einer Uhr nummeriert, das Zentrum ist (0|0):

  • Koordinaten eines liegenden Sechsecks

P1 (4|7), P3 (8|0), P5 (4|−7), P7 (−4|−7), P9 (−8|0), P11 (−4|7)

  • Koordinaten eines auf der Spitze stehenden Sechsecks
P2 (7|4), P4 (7|−4), P6 (0|−8), P8 (−7|−4), P10 (−7|4), P12 (0|8)


English: The name of the graphic says that it is possible to position only by counting computation of square boxes 12 points on a circular path so that it results in an almost regular dodecagon.
approximation at 30° with a tangent of 7/4
Deviation from the ideal < 0.8%

For many practical purposes, you can reach a sufficient approximation to a regular hexagon when you designed this for nearly equilateral triangles. A nearly equilateral triangle is obtained when the base of the triangle to the amount present is in a ratio of 8:7. The calculation in the graph shows that the two legs of the triangle thus constructed have nearly the length of the base side. This approach is used in applications where an error by almost 1% can be tolerated . Thus, for example, cards or playing fields and easy to drawings can be divided easily be calculated sectors. The design also easily manage children of primary school age, there is not more needed, as the counting of computational box. Elaborate constructions with compass and set square is no longer necessary.


Practical advantage: simple design and graphics, cartography, construction of hexagons on the computer by drawing programs without angular function (only with points). For larger structures could also be interesting to have a ratio of 7:12

72 + 122 = 193 = 13,89...2 ~ 196 = 142.

Deviation from the ideal < 0.8%

vector coordinates

The points Pi (x|y) depend on the numbers of a clock, the center is (0|0):

  • Coordinates of a hexagon is
P1 (4|7), P3 (8|0), P5 (4|−7), P7 (−4|−7), P9 (−8|0), P11 (−4|7)
  • Coordinates of a hexagon below the top
P2 (7|4), P4 (7|−4), P6 (0|−8), P8 (−7|−4), P10 (−7|4), P12 (0|8)
Annotations
InfoField
This image is annotated: View the annotations at Commons

File history

Click on a date/time to view the file as it appeared at that time.

Date/TimeThumbnailDimensionsUserComment
current11:31, 11 December 2011Thumbnail for version as of 11:31, 11 December 2011400 × 400 (59 KB)Mjchael (talk | contribs){{Information |Description= Square of the circle (approximation) |Source = made by me (Inkscape) |Date = 2007-08 |Author=user:Mjchael |Permission = {{self|cc-by-sa-2.5}} }} ;Deutsch Der Name der Grafik besagt, dass es möglich ist, nur durch das Abz

You cannot overwrite this file.

The following page uses this file:

File usage on other wikis

The following other wikis use this file:

Retrieved from "https://commons.wikimedia.org/w/index.php?title=File:Construction_of_an_almost_regular_hexagon_and_dodecagon..svg&oldid=867728528"