Enneagram (geometry)

{{Short description|Nine-pointed star polygon}} {{About|the geometric polygon|other uses|Enneagram (disambiguation)}} {{Redirect|Nonagram|the puzzle|Nonogram}} {{Infobox Polygon | name = Enneagram | image = Enneagon stellations.svg | caption = Enneagrams shown as sequential stellations | edges = 9 | schläfli = | coxeter = | symmetry = Dihedral (D9) | area = | angle = 100° {9/2} 20° {9/4} | properties =}} {{Star polygons}} In geometry, an '''enneagram''' (🟙 U+1F7D9) is a nine-pointed plane figure. It is sometimes called a '''nonagram''', '''nonangle''', or '''enneagon'''.<ref>{{Cite web|url=http://chalkdustmagazine.com/blog/fractional-polygons/|title=Between a square rock and a hard pentagon: Fractional polygons|date=28 September 2017}}</ref>

The word 'enneagram' combines the numeral prefix ''ennea-'' with the Greek suffix ''-gram''. The ''gram'' suffix derives from ''γραμμῆ'' (''grammē'') meaning a line.<ref>[https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3Dgrammh%2F γραμμή], Henry George Liddell, Robert Scott, ''A Greek-English Lexicon'', on Perseus.</ref>

==Regular enneagram==

A '''regular enneagram''' is a 9-sided star polygon. It is constructed using the same points as the regular enneagon, but the points are connected in fixed steps. Two forms of regular enneagram exist: *One form connects every second point and is represented by the Schläfli symbol {9/2}. *The other form connects every fourth point and is represented by the Schläfli symbol {9/4}.

There is also a star figure, {9/3} or 3{3}, made from the regular enneagon points but connected as a compound of three equilateral triangles.<ref>Grünbaum, B. and G. C. Shephard; ''Tilings and patterns'', New York: W. H. Freeman & Co., (1987), {{ISBN|0-7167-1193-1}}.</ref><ref>Grünbaum, B.; Polyhedra with Hollow Faces, ''Proc of NATO-ASI Conference on Polytopes ... etc. (Toronto 1993)'', ed T. Bisztriczky et al., Kluwer Academic (1994) pp. 43-70.</ref> (If the triangles are alternately interlaced, this results in a Brunnian link.) This star figure is sometimes known as the ''star of Goliath'', after {6/2} or 2{3}, the star of David.<ref>{{Cite web|url=https://mathworld.wolfram.com/Nonagram.html|title=Nonagram|first=Eric W.|last=Weisstein|authorlink=Eric W. Weisstein|website=mathworld.wolfram.com}}</ref>

{| class=wikitable !Compound !Regular star !Regular compound !Regular star |- align=center |120px Complete graph K9 |120px {9/2} |120px {9/3} or 3{3} |120px {9/4} |}

==Other enneagram figures== {| width="480" |- valign=top |150px The final stellation of the icosahedron has 2-isogonal enneagram faces. It is a ''9/4'' wound star polyhedron, but the vertices are not equally spaced. |150px The Fourth Way teachings and the Enneagram of Personality use an irregular enneagram consisting of an equilateral triangle and an irregular hexagram based on 142857. |150px The Baháʼí nine-pointed star |150px A ''9/3'' enneagram |150px The star of Eldia from ''Attack on Titan'' |} The nine-pointed star or enneagram can also symbolize the nine gifts or fruits of the Holy Spirit.<ref>''Our Christian Symbols'' by Friedrich Rest (1954), {{ISBN|0-8298-0099-9}}, page 13.</ref>

==In popular culture== *The heavy metal band Slipknot previously used the {9/3} star figure enneagram<ref>{{Cite web|url=https://www.ebay.com/sch/i.html/main.php/i.html?_from=R40&_nkw=slipknot&_sacat=0&_trksid=p2510209.m570.l1313&sk=nonagram|title=slipknot|website=eBay}}</ref> and currently uses the {9/4} polygon as a symbol. The prior figure can be seen on the cover of their album ''All Hope Is Gone''. *The symbol of Eldia from ''Attack on Titan'' is an irregular enneagram, representing the nine Titan powers.

==References== {{Reflist}}

'''Bibliography''' *John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, ''The Symmetries of Things'' 2008, {{ISBN|978-1-56881-220-5}} (Chapter 26. pp. 404: Regular star-polytopes Dimension 2)

==External links== *{{Commons category-inline}} *[http://mathworld.wolfram.com/Nonagram.html Nonagram -- from Wolfram MathWorld]

Category:9 (number) 09