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Pentagram

A pentagram is a five-pointed star polygon with deep roots in geometry, religion, magic and art. This article explains its form, historical development, meanings and common distinctions such as the pentacle.

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Overview

The pentagram is a five-pointed star formed by drawing five straight lines that connect five equally spaced points on a circle so that the lines intersect. It is one of the simplest star polygons and appears across cultures as a decorative motif, a mathematical figure and a symbol with religious or magical associations. Alternative names include pentangle, pentalpha and star pentagon.

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Geometry and construction

Geometrically the pentagram is closely tied to the regular pentagon. When a pentagram is drawn inside a regular pentagon, the crossings generate smaller similar pentagons and a network of segments whose proportions relate to the golden ratio. Many classical constructions use straightedge-and-compass methods to create the pentagram from a regular pentagon or from five points on a circle. For geometric details and angle relationships see related geometry resources. The pentagram can be considered a simple {5/2} star polygon in Schläfli notation and is often studied in elementary plane geometry.

History and development

The symbol appears in ancient art and inscriptions from several regions, including early Mesopotamia and Greece, and was noted by classical writers. In the classical period it attracted mathematical interest—Pythagoreans valued the pentagram as a sign of harmony and proportion. During the medieval and Renaissance eras the pentagram was used in Christian iconography and in treatises on natural philosophy. By the early modern period it also became associated with ceremonial magic and alchemy. Over time the same basic figure acquired many different and sometimes opposing meanings.

Uses, meanings and examples

The pentagram serves diverse roles depending on context:

  • Religious and spiritual: It has been used as a symbol of protection, of the five wounds of Christ in some Christian contexts, and as a talisman in various magical traditions.
  • Occult and modern Paganism: In contemporary Wicca and related paths a pentagram often represents elements or spirit; a pentagram enclosed in a circle is commonly called a pentacle. For information on modern practice see resources about Wicca.
  • Mathematical and artistic: Artists and architects use pentagram motifs for decoration; mathematicians study its proportions and self-similar structure, often linking it to the golden ratio and pentagonal symmetry (see pentagon).

Distinctions and modern associations

Two distinctions are frequently noted. First, a pentacle is often defined in modern usage as a pentagram circumscribed by a circle and used as an amulet or ritual object; historically the word had a broader meaning referring to protective symbols. Second, orientation affects interpretation: an upright pentagram (single point up) is commonly used to symbolize harmony or spirit above matter, while an inverted pentagram (two points up) has been adopted in some 19th–20th century contexts as a symbol of inversion or opposition and is sometimes associated with anti-religious or Satanic imagery. These associations are culturally contingent and have changed over time.

Notable facts

Because it recurs across separate traditions, the pentagram is a useful example of how a simple geometric figure can carry many layers of meaning. Its mathematical properties—self-similarity, connections to the golden ratio and pentagonal symmetry—explain in part its aesthetic appeal. Whether encountered in ancient inscriptions, medieval manuscripts, modern jewelry or mathematical diagrams, the pentagram remains a compact symbol that bridges art, science and belief.

Pentagram,Schläfli symbol {5/2}, {5/3}.
Pentagram,Schläfli symbol {5/2}, {5/3}.

Unicode

The pentagram is included in the Unicode block Miscellaneous Symbols. At code point U+26E4 (9956) ⛤ under the name "PENTAGRAM" and at code point U+26E7 (9959) ⛧ as "INVERTED PENTAGRAM".

Geometry

Construction

→ Main article: Pentagon

From five points distributed evenly, i.e. at intervals of 72°, on a circle, two five-axis symmetrical figures can be created by means of chords:

  • The five chords between adjacent points form a regular pentagon with angles of 108° each.
  • The five chords between non-adjacent points together form the pentagram with acute angles of 36°.

Geometric properties

The inner sections of the chords of the pentagram again form a regular pentagon. Compared to the outer it is rotated by 36°. The prongs of the pentagram are isosceles triangles. The angles between the base and the legs of these triangles are 72°.

The inner pentagon, together with two non-adjacent prongs each, forms an isosceles triangle with an obtuse apex, the 108° angle already mentioned.

If we draw another pentagram in the inner pentagon, its chords together with parts of the chords of the outer pentagram also form isosceles triangles with an obtuse apex of 108°. Their median perpendiculars are parallel to those of the pentagonal triangles, which in turn form the axes of symmetry of all pentagrams and pentagons.

Outer pentagon, pentagram and inner pentagon have the same center. Each chord of the inner pentagram is parallel to a chord of the outer pentagram located beyond the center.

All angles between the edges of the pentagram and the enclosing pentagon are therefore 36°, 72° or 108°. The five axes of symmetry have angles to the edges of 18°, 54° and 90°.

golden section

All chords and chordal parts of a pentagram bounded by intersections, including the outer and inner pentagons, have only four different lengths. Of these, each successive one is in the ratio of the golden section, i.e. the following length ratios are equal:

{\displaystyle {\frac {AB}{BC}}={\frac {AC}{AB}}={\frac {AD}{AC}}}

For example, it applies to the first ratio:

{\displaystyle CC'=CD}

The triangle {\displaystyle CC'D} is isosceles because it is a proportional diminution of the isosceles triangle , {\displaystyle BB''D}since the interior angles of the two triangles are equal.

{\displaystyle AB=CD}and {\displaystyle BB'=BC}

Due to the ray theorem holds:

\frac{AB}{BB'} = \frac{AC}{CC'}

Consider the relationship

{\displaystyle \Phi ={\frac {AB}{BC}}={\frac {AC}{AB}}={\frac {AB+BC}{AB}}},

then {\displaystyle \Phi =1+{\frac {1}{\Phi }}}and thus satisfies the definition of the golden section, thus {\displaystyle \Phi ={\frac {1+{\sqrt {5}}}{2}}=1{,}618...}.

Formulas

Mathematical formulas for the pentagram

Area

{\displaystyle A={\frac {1}{4}}\cdot {\sqrt {250+110\cdot {\sqrt {5}}}}\cdot s^{2}\approx 5{,}568\cdot s^{2}}

{\displaystyle A={\frac {1}{2}}\cdot {\sqrt {25-10\cdot {\sqrt {5}}}}\cdot a^{2}\approx 0{,}812\cdot a^{2}}

Scope

{\displaystyle U=5\cdot l={\frac {5+5\cdot {\sqrt {5}}}{2}}\cdot a=(10+5\cdot {\sqrt {5}})\cdot s}

Tendon length

{\displaystyle l=\Phi \cdot a={\frac {1+{\sqrt {5}}}{2}}\cdot a\approx 1{,}618\cdot a}

{\displaystyle l=\Phi ^{3}\cdot s=(2+{\sqrt {5}})\cdot s\approx 4{,}236\cdot s}

{\displaystyle l={\frac {1}{2}}\cdot {\sqrt {10+2\cdot {\sqrt {5}}}}\cdot r\approx 1{,}902\cdot r}

Side length of the inner pentagon

{\displaystyle s={\frac {a}{\Phi ^{2}}}={\frac {3-{\sqrt {5}}}{2}}\cdot a\approx 0{,}382\cdot a}

{\displaystyle s={\frac {l}{\Phi ^{3}}}=({\sqrt {5}}-2)\cdot l\approx 0{,}236\cdot l}

{\displaystyle s={\frac {1}{2}}\cdot {\sqrt {50-22\cdot {\sqrt {5}}}}\cdot r\approx 0{,}449\cdot r}

Aspect length of the outer pentagon

{\displaystyle a=(\Phi +1)\cdot s=\Phi ^{2}\cdot s={\frac {3+{\sqrt {5}}}{2}}\cdot s\approx 2{,}618\cdot s}

{\displaystyle a=(\Phi -1)\cdot l={\frac {l}{\Phi }}={\frac {{\sqrt {5}}-1}{2}}\cdot l\approx 0{,}618\cdot l}

{\displaystyle a={\frac {1}{2}}\cdot {\sqrt {10-2\cdot {\sqrt {5}}}}\cdot r\approx 1{,}176\cdot r}

Perimeter radius

{\displaystyle r={\frac {1}{10}}\cdot {\sqrt {50-10\cdot {\sqrt {5}}}}\cdot l\approx 0{,}526\cdot l}

{\displaystyle r={\frac {1}{10}}\cdot {\sqrt {50+10\cdot {\sqrt {5}}}}\cdot a\approx 0{,}851\cdot a}

{\displaystyle r={\frac {1}{10}}\cdot {\sqrt {250+110\cdot {\sqrt {5}}}}\cdot s\approx 2{,}227\cdot s}

Polyhedron with pentagrams

Some polyhedra have pentagrams as side faces, for example the dodecahedron star and the icosahedron star. These two polyhedra are Kepler-Poinsot solids.

·        

Dodecahedron star

·        

Icosahedron Star

The points A, B, C, D, E define a regular pentagon (red) and a pentagram (violet)
The points A, B, C, D, E define a regular pentagon (red) and a pentagram (violet)
A fractal and self-similar pentagram
A fractal and self-similar pentagram
Pentagram
Pentagram

Questions and answers

Q: What is a pentagram?

A: A pentagram is a five-pointed star, with all lines the same length and all angles the same.

Q: What other names are used to refer to a pentagram?

A: A pentangle, star pentagon, or pentalpha means the same thing as pentagram.

Q: What was the original meaning of the word pentacle?

A: The word pentacle originally meant 'any symbol that protects against evil spirits'.

Q: What is the second definition of 'pentacle' in modern Wicca?

A: The second definition of 'pentacle' in modern Wicca is 'a circumscribed pentagram', which means 'a pentagram drawn inside a circle so that the points of the pentagram touch the circle'.

Q: What is the significance of a pentagram in Wicca?

A: A pentagram is used as a symbol of protection and is a key symbol in Wicca.

Q: How have the meanings of pentagrams changed over time?

A: The meanings of pentagrams have changed over time and can vary from culture to culture. In modern Wicca, the pentagram is seen as a symbol of protection and spirituality.

Q: Are there other symbols that protect against evil spirits?

A: Yes, there are other symbols that protect against evil spirits. The pentagram is just one of many such symbols.

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