What is a Star?
The star above is a regular hexagon with six equilateral triangles on its
||If you look at an encyclopedia, you'll find several meanings of the
>"a heavenly body"
>"something that sends rays to all sides"
>"a new famous person"
>a "thing with regular points around". (Brockhaus 1975). - I mean this
You can define: A star is a figure which generally consists of a convex
polygon with triangles on its sides.
The triangles are called points.
Producing of Stars top
There are different ways of making a star.
You take a regular polygon and set equilateral
triangles on the sides.
This procedure doesn't lead to a star, if you start with a triangle.
You take a regular polygon and reflect
the basic triangles of the polygons on a side.
This procedure doesn't lead to a star, if you start with a triangle or
You take a regular polygon with n corners and
turn it around 180/n. Origin and image together form a star.
Give a polygon and draw special diagonals. They form stars. Regular
polygons form stars with rich symmetries. The pentagram and the hexgram
are the simlest examples.
You take a regular polygon and set isosceles
triangles on the sides, which have a large height compared with the basic
side. The polygon, which has the points, can degenerate into a circle (far
Handsome stars develop. Obviously you expect that stars must have "pointed
You take any polygon and set arbitrary
triangles on the sides.
||The star of the German magazine "Stern" was a model of this star.
Making of Stars top
The star is a favourite topic for many people
doing handicrafts with innumerable variations. I introduce the following
Star as a silhouette
The first eight pointed star
is a silhouette.
This star can variously be modified. You don't only cut out a wedge, but
make other forms and leave bars. You can also use black paper and close
the holes with coulored paper. Then window pictures develop.
||You fold a square sheet of paper again and again along the red line,
until fig.5 develops. Then you cut off pieces along the blue lines
with strong kitchen scissors. Fig.6 develops. While unfolding you get the
The second star is Froebel's star, which is made
from four strips. This is explained on another site of my homepage.
The Pentagram certainly is "the star" among the stars. It is also called
Drudenfuß (German) or witch star.
Which angles appear?
||It develops if you draw the diagonals (black) inside a regular pentagram.
||The central angle (blue) in the central triangle ABC of the regular
pentagon is 360°/5 = 72°.
So the two base angles are 54°. The interior angle of the pentagon
||Assuming that the small pentagon inside the pentagram also is regular,
the marked orange angle is 108°. The point is an isosceles
There is a second smaller yellow 72-36-72-triangle lying at the bottom.
||Both yellow triangles are similar to each other because of the corresponding
You'll find the formulas in the next chapter.
You can say: In a regular pentagon the diagonals cut each other in a golden
ratio. You can find the ratio from the proportion a:x=x:(a-x). The product
equation is a(a x)=x², so x²+ax-a²=0. If you solve this
quadratic equation, you get the positive solution x:a =1/2*sqr(5)-1/2.
This is about 0.62.
||It is PQ:QS=QS:QR or a:x=x:(a-x) in the
yellow triangles. This equation can be related to the line PQ and the point
R on the line. The whole distance is to the larger section as this larger
section to the smaller one. You can also say: The larger section is the
(golden) mean of the whole and the smaller section, x=sqr[a(a-x)]
The well known German TV journalist Professor Heinz
Haber described the golden ratio as follows in the 1960s.
||There are a side board and a vase with flowers in an experimental room.
Test subjects had to put the vase on the cabinet in such a way so that
it looks nice.
Only a few people placed it in the centre. That looks boring. Most people
put the vase slightly on the right or on the left as shown in the picture.
The result was that the width of the cabinet was divided in the golden
ratio. Perhaps this sense of beauty is given by education and belongs to
our western culture.
The pentagram is called also Drudenfuss or witch
star. This is a reference to a certain topic you can investigate on the
internet. I recommend the picture directory of google.com and the search
word "pentagram". Sometimes you think you are in the Middle Ages. (This
is the same with the hexagram in the next chapter.)
My contribution can be: I saw a pentagram at the door of the combined
horse/cow stable of a farm in the 1950s. It should keep away diseases from
||You can draw the pentagram or stars with 7, 9... points with free hand
after some practise without lifting the pencil.
I impressed many people by this :-).
A second star worth being described is the hexagram. It is also called
David's star or Salomon's seal.
||The hexagram consists of 12 equilateral triangles and therefore belongs
to the Polyiamonds.
||You can also regard the hexagram as a figure, which is formed by two
equilateral triangles. One triangle stands on its head.
In former times Salmiakpastillen (bitter-tasting lozenge) were in fashion.
You moistened the (clean?) back of your hand and form a star from six pastilles
on it. Then you licked the pastilles.
Two of Karen Koeller's
Three-dimensional Stars top
You can design 3d stars of regular solids. You put a pyramid on each
Here is an octahedron with eight tetrahedra as an example.
Those who know the 3d view can see the star three-dimensional.
The Kepler-Poinsot Polyhedra
||If you put some regular pentagonal pyramids on the sides of a
dodecahedron and tetrahedra on a icosahedron, you get the "Small Stellated
Dodecahedron" and the "Great Stellated Dodecahedron" (Kepler). Poinsot
discovered the two "Poinsot polyhedra", the "Great Dodecahedron"
and the "Great Icosahedron".
The Great Dodecahedron is well known as Alexander's Star (on the left),
a puzzle belonging to the Rubik's cube family.
You find more about these interesting solids on the internet pages below.
There are three-dimensional stars, which hang in
the hall of many houses in December. They have a bulb in the centre so
they shine from inside. They are called Sternlampions (=star Chinese lanterns)
in German and Moravian stars in English. Among them the "Original Herrnhuter
Weihnachtsstern" (Herrnhut Star) is known all over the world.
Stars - Near and Far Away top
Many nations have one or several stars in their national flag. Here
is a small choice.
Here in Lippe, a district of North Rhine-Westphalia there are
many half-timbered houses with a big hall, so that horse and carriage could
drive in in former times. Usually there is an inscription outside above
the hall with the star (here seven-pointed) of "Burg Sternberg" on one
side and on the other side the "lippische Rose" similar to a star.
House of "Gut Hovedissen" at Leopoldshöhe, Kreis Lippe, as an example.
A curiosity is in the coat of arms of the town Barntrup/Lippe (on the right).
"Der halbe Stern erinnert an die Gründung der Stadt durch die Grafen
zu Sternberg. Die halbe lippische Rose wurde von der Stadt nach dem Übergang
der Grafschaft Sternberg an die Herrschaft Lippe in das städtische
You also find the star and the rose at the historic
town hall (1546) of my hometown Bad Salzuflen.
||The Lippische Rose is also (although small) in the flag of the State
This was a consequence of the connection of the state Lippe to NRW after
World War II.
Stars on Other Places of my Homepage
Stars on the Internet top
Birgit Bachmann und Stefan R. Müller (Blinde Kuh)
(u.a. Sterne aus Papier, Siebenzackiger Stern, Schneesterne, Drachensterne,
Faltsterne, Plastischer vierzackiger Stern, Einfacher Faltstern)
Das Hexagramm, .pdf-Datei
Der Herrnhuter Stern
Die stetige Teilung
oder der goldene Schnitt und die Konstruktion des regelmäßigen
die Pythagoreer und der Goldene Schnitt
How to Cut a
5-Pointed Star in One Snip
Eric W. Weisstein (World of Mathematics)
Harvey D. Heinz
Pastimes [ACTIVITY LINKS, ACTIVITY 15 (Stellations of Regular Polyhedra)]
Suzanne Alejandre and Mutsumi Suzuki
(1) Martin Gardner: Mathematischer Zirkus, Berlin 1988 (ISBN 3550076924)
Feedback: Email address on my main page
page is also available in German.
2003 Jürgen Köller