The number five plays an important role in determining the value of the golden ratio (Meisner, 2012). Phi can be derived from the following formulas shown below. It becomes apparent from looking at these, that the number five is an essential element in determining phi. Not only does the number five feature in the formula for the golden ratio but it is also a number in the Fibonacci sequence (Scott and Gulick, 2010, p. 38).

When analysing the number five in a geometrical context by looking at a regular pentagon, the golden ratio appears in many different ways (Darvas, 2007, p.173). Below I have created a diagram of a regular pentagon, showing various segments that relate to the golden section.

Regular Pentagon

The diagonals and segments constructed from the vertices of the pentagon represent the value of phi in many different ways. For example, the line segment AB divided by segment BX gives the golden ratio. Dividing segment BV by BX also equal to the value of phi. All proportion of line segments shown below are equal to the golden ratio or phi.

Proportions of the regular pentagon have influenced various art, particularly works relating to the proportions of the human body (Darvas, 2007, p.175). The regular pentagon was used by Leonardo da Vinci, for example, to create proportions of the ideal human body. In the figure of the Vitruvian Man he placed limbs at the vertices of the regular pentagon (Stakhov and Olsen, 2009, p.43).

‘Vitruvian Man’ by Leonardo da Vinci

Lesson Ideas for Pentagon Topic

• Examining pieces of art that that use pentagons (link to tessellations)
• Why don’t pentagons tessellate?
• Constructing a five pointed star using a regular pentagon
• Regular and irregular pentagons

References

Darvas, G. (2007) Symmetry: Cultural-historical and Ontological Aspects of Science-Arts Relations. Basel: Birkhäuser Verlag.

Image. Photograph. Available at: http://109.74.193.65/dev/wordpress/wp-content/uploads/vitruvio01-400×562.jpg (Accessed: 20 November 2015)

Scott, J. and Gulick, D. (2010) The Beauty of Fractals: Six Different Views. Washington, D.C.: Mathematical Association of America.

Meisner, G. (2012) ‘Mathematics of Phi, 1.618, the Golden Number’, Golden Number, 16 May. Available at: http://www.goldennumber.net/math/ (Accessed: 21 November 2015)

Scott, J. and Gulick, D. (2010) The Beauty of Fractals: Six Different Views. Washington, D.C.: Mathematical Association of America.

Stakhov, A. and Olsen, S. (2009) The Mathematics of Harmony: From Euclid to Contemporary Mathematics and Computer Science. Singapore: World Scientific Publishing Company.