the golden ratio | My Education ePortfolio

I came out of this lecture completely mind blown at our world and what is produces naturally, we truly live in an extraordinary reality. Today we learnt about the Fibonacci sequence (the golden spiral) together with Phi (the golden ratio).

In 1509 there was an Italian mathematician called Luna Pacioli who published Divina Proportione, which was a treatise on a number that is known as the ‘Golden Ratio’. We symbolise this ratio by Phi (Φ). This ratio comes with fascinating frequency in nature all around us and mathematics (Pickover, 2009).

The Golden Spiral is made up of the Fibonacci sequence. The sequence is made by the fact that every number after the first two is the sum of the two preceding ones. The Rule is x_n = x_n-1 + x_n-2. The sequence goes:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, … to infinity.

If you draw these numbers out in length x breadth boxes then it creates the golden spiral.

This spiral can be seen in a vast amount of nature from our galaxies to the shell of the nautilus. It is truly mind blowing to say the least, the fractal nature of reality.

A fractal is a way of seeing infinity – Benoît Mandelbrot

The natural world has a fondness of the Fibonacci numbers. If you look at flowers most of them have a Fibonacci number of petals.

3 petals = lily & iris

5 petals= buttercups

8 petals = delphinium

13 petals = marigold & ragwort

21 petals = aster

55/89 petals = daisy

Not all flowers will have these numbers but averagely they do. For instance this is why 4 leaf clovers are so rare as the number 4 is not in the Fibonacci sequence (Bellos, 2010).

Just as pi (π) stands for the ratio of the circumference to the diameter of a circle, Phi (Φ) stands for a special ratio of line segments. When a line is divided in a unique way the ratio Phi happens. “We divide a line into two segments so that the ratio of the whole segment to the longer part is the same as the ratio of the longer part to the shorter part” (Pickover, 2009, p.112).

(a + b)/ b = b/a

The ratio (golden ratio) is 1.61803….

Using this ratio Anna asked us to work in pairs to see how “beautiful” our bodies were. By dividing different measurements by each other we were able to calculate, if our body part = 1.6… then they were “beautiful”. Me and Ellie Kean calculated we both had “beautiful” heights.

The Greeks were amazed by this ‘phi’. They founded the 5-pointed star, it was the admired symbol of the Pythagorean Brotherhood. It was called the ‘extreme and mean ratio’ by Euclid and he was able to make it by a compass and a straightedge method (Bellos, 2010).

Leonardo da Vinci is a prime example of an artist who believed maths and art has a strong bond. This is clearly seen in his most famous drawing of the ‘Vitruvian Man’. The drawing shows mathematically and artistically that the human body is has its perfectly symmetrical measurements and dimensions not by coincidence.

Learn how to see. Realise that everything connects to everything else – Leonardo da Vinci

Overall, there is a true connectedness of mathematics and art. There is proof of this in our worlds nature make-up and has been discovered through history also with the help of Leonardo da Vinci. I can use Leonardo as an inspiration within my future art lessons with students so they can have a more broad understanding of the history of art and how it connects to art we see now in the present day.

References:

Pickover, C. A. (2009) The Math Book From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics. London: Sterling.

Bello, Alex (2010) Alex’s Adventures in Numberland London: Bloomsbury

Up until this point in my education I’ve always thought of maths as a subject full of equations, difficult strategies and complicated rules. Especially in higher maths at school is was a constant routine of go to class, be taught a new subject for half the lesson with vigorous note taking and then the second half of the lesson would be textbook work. To make matters worse if you didn’t finish the textbook work you had to do it as additional homework…so from my point of view there was very little if not no creativity for the most part of my mathematical education in high school. The early years of primary school is probably where creative maths would be found, but I know what I learnt in my lecture today couldn’t be taught to a 6 year old and I think the more advanced maths is better used with creativity as it’s more of an “ah ha” moment.

In my Discovering Maths lecture today I was able to see how creativity is in maths all around us. Artists throughout history have used such maths topics as symmetry, tessellation and proportion. Ancient Greek architects and sculptors used the golden ratio (I’ll speak more about this in another blog!) to make sure that buildings were physically nice to look at. Portrait painters form the renaissance period made sure that proportions of their portraits facial features and head were in proportion to the rest of their body, to do this they followed mathematical ratios. Tessellations and geometric shapes are seen in Islamic art and represents spirituality of the world.

What are tessellations you ask? A tessellation is a pattern of shapes that fit perfectly together, with no gaps, and create a pattern that could repeat for infinity and beyond.

Islamic art avoids the use of human figures or animals in their art. But rather has three principle elements which they include in each piece. These are:

Calligraphy
Arabesque
Geometry

Sacred Geometry gives symbolic meanings to types of geometric proportions/shapes. it is also associated with the belief that “god is the geometer of the world” (Sacred geometry, 2017). Geometry is said to be at the heart of nature, it is seen in all of nature if you look close enough, and therefore makes us all develop so beautifully. And so it is at the heart of Islamic art. “In the world of natural phenomena, it is the underlying patterns of geometric form, proportion and associated wave frequencies that give rise to all perceptions and identifications. Therein lies our fundamental capacity to relate, to interpret and to know.” (Bansal, 2014). Aspects of nature that geometry can be seen in is honeycombs, spirals in flowers, pine cones, foliage on trees etc. Nature’s design links to the golden ratio, divine proportion, phi and consciousness. The chambered nautilus grows constantly and its shell creates a logarithmic spiral shape and hold the growth without the shell changing its shape, this shape links to the golden ratio which we will learn more about later on (Crystal, no date).

‘The seed of life’ is a concept relating to flower patterns. Within the flowers geometry we start with the seed which includes 6 circles interlinked together which creates with perfect pattern that you may see used in company logos. Once the see germinates it starts to turn into a flower and the geometry continues by adding a further 6 circles, turning the pattern into 12 interlinked circles. This is an fascinating design from natures basic friend, the flower. These patterns are universal in nature and if you were to go to another universe or planet you’d find similar if not the same pattern there too! (McConaghay, 2016).

Within Islamic art there are three fundamental shapes used:

Equilateral triangle –

This is the simplest shape to draw and fit together and it represents harmony and human consciousness.

Square –

Also an easy shape that fits together and it represents the four corners of the earth.

Hexagon –

This shape represents heaven.

The star whether it is a 6, 8, 10 or 12 pointed star is also often used in Islamic art.

From this lecture/workshop we got to be fully creative with paints, coloured paper and a load of cutting out (being safe with the scissors at all times!). We made our own versions of tessellations which all turned out very pretty if i may say so myself.

Me, myself am a very creative person in general. I dance, play a musical instrument and took art all the way up to advanced higher at school, therefore my classroom in the future won’t be short of creativity and a splash of colour and art in all aspects of learning!

References:

McConaghay, D. 2016. [Website] Available at: https://www.gaia.com/article/sacred-geometry-nature. (Accessed 02/11/17).

Crystal, E. no date. [Website] Available at: http://www.crystalinks.com/sg.html. (Accessed 02/11/17).

Bansal, A. 2014. [Website] Available at: http://www.archinomy.com/case-studies/1938/geometry-nature-architecture. (Accessed 02/11/17).

Sacred Geometry. (2017). [Website] Available at: https://en.wikipedia.org/wiki/Sacred_geometry. (Accessed 02/11/17).

🌻 The Fractal Nature Of Reality 🍀

Maths, Creative? No Way! 🎨