The Genius of James Clerk Maxwell and the Theory of Colours Oscar Burke

The genius of James Clerk Maxwell and the Theory of Colours Oscar Burke

Figure 1: Statue of James Clerk Maxwell, George Street, Edinburgh. 13

In Nature I read Quite a different creed, There everything lives in the rest; Each feels the same force, As it moves in its course, And all by one blessing are blest. The end that we live for is single, But we labour not therefore alone, For together we feel how by wheel within wheel, We are helped by a force not our own So we flee not the world and its dangers, For He that has made it is wise, He knows we are pilgrims and strangers, And He will enlighten our eyes. (James Clerk Maxwell, 1858, Tune, Il Segreto Per Esser Felice, III.) Introduction James Clerk Maxwell is said to be one of the greatest minds in physics, rivalling that of Isaac Newton and Albert Einstein; curiously, however, his name is not a house hold name. His work inspired physicist Albert Einstein to say that James Clerk Maxwell’s contributions to physics “changed the world forever.” German physicist Max Plank, who found that energy came in little packets called quanta, said Maxwell “achieved greatness unequalled.” American astronomer and narrator of the popular science documentary show Cosmos Carl Sagan, said “Maxwell's equations have had a greater impact on human history than any ten presidents.” So who is James Clerk Maxwell and what is all the fuss about? Figure 2: James Clerk Maxwell with his James was born on the 13th of June 1831 in Edinburgh at 14 colour top - which led him to confirm Young's theory of colour7 India Street; James was born into a loving and esteemed family. His mother, Frances, died in 1839, which formed a very close relationship with his father, John, who was a lawyer. He started schooling at Edinburgh Academy, an elite school where his country attire of square shoes and being dull in class gained him the name “Dafty”. However, James was notorious for being good-humoured and was able to ferociously defend himself, much to the surprise of his bullies in the school yard. He also showed strong academic promise, questioning things far beyond the people of his age around him. It was here that his father John decided to get him a tutor, who used traditional means of punishment like the cane to ‘help’ James understand each lesson. This torment got to such a point that James ran down to the lake with a large washing tub and paddled out to the middle of the lake, beyond the reach of his violent tutor! James found Edinburgh Academy rather dull, but at home he read and made his own discoveries, and with little formal knowledge of geometry created a tetrahedron, dodecahedron and two others ‘he did not know the name of’. It was his Figure 3: Plaque with Maxwell's Equations demonstrating how friendship with Lewis Campbell that put electricity and magnetism are linked together an end to his isolation at school, Lewis (By Lourakis - Own work, CC BY-SA 4.0, - Wikimedia had a similar inquisitive mind to James and their chatter was endless full of thought- provoking ideas. At fourteen, James had produced and had published his first paper on oval curves (which he made using pins and string). His father John showed James’ solution to James Forbes, a mathematics professor at Edinburgh University. To his surprise, James had found a more general and less complex solution than the solution by the famous philosopher and mathematician Rene Descartes in the seventeenth century. It was here that James got his first taste of academia. James went to discover that electromagnetic radiation moved so close to the speed of light that it must also be light (this theory was confirmed a quarter of a century later by German physicist, Heinrich Hertz!). He was the first to show how to calculate stresses in suspension bridges. He predicted why there were gaps in between the rings of Saturn, and he helped to create the world’s first colour photograph (Figure 3). His work “A Dynamical Theory of the Electromagnetic Field” helped Einstein create the theory of General Relativity in 1915 and has shown the mathematics, known as “Maxwell’s Equations” (figure 3), behind electromagnetic radiation. Applications of Figure 4: The world’s first colour photograph – a picture of a tartan which have helped develop radio, television, ribbon (no one could repeat this experiment as the dyes to make the photos back then did not work. 100 years later, Kodak discovered radar, microwaves and thermal imaging. that the ribbon reflected ultraviolet light (purple UV light) as well as red, which exposed the film where the red should have been! Nice This paper immerses students in accident! (Illustration: Photograph: Thomas Sutton/ James Clerk mathematics through simulating the colour Maxwell/ SSPL - Getty Images) top experiment, which led Maxwell to confirm how we perceive colours and how it was used to diagnose colour-blind people and to produce the world’s first colour photograph.8 History Maxwell introduced to colour Maxwell was an intensely curious boy, finding lots of different uses for regular objects. Bored of using a spoon as spoons ought to be used, he was found using the back of the spoon to bring the sun into the house. When told about a blue rock, he answered “How d’ye know it’s blue?”.5 A question that we still find troubling when asked today. He carried this burning curiosity along with his fascination with colours, with him to his adulthood. Young, a famous physicist, thought that the human eye has only three types of receptors, which he couldn’t prove. James Forbes, James Maxwell’s professor at Glasgow Figure 5: Diagram of Maxwell's Coloured top. An inner circle and outer circle of University, was experimenting with this problem himself. coloured disks. The amount that each disk He found that by placing and overlapping multiple pieces of could be exposed could be varied. 6 circular coloured paper (so that the paper looked like a pie chart) and then spinning the top produces some very interesting results. He found that the colours on the spinning top began to merge together into one colour when the top was spinning at sufficient speed! Merging colours and blurring were nothing new, there were a wide range of popular optical toys that were available in the 1830s.3 However, what baffled Forbes was that the single colour that the spinning disks were making did not match the colour that was expected. Forbes knew from painters and dye makers that three colours; being red, yellow and blue; were required to make paint or dye of any other colour in the visible spectrum! Forbes tried blue and yellow pieces of paper together (which according to the painters and dye-makers should make green) and he instead produced a dull pink! With other projects and duties to occupy him, Forbes gave young James Maxwell free reign in his laboratory. Maxwell wanted to solve Young’s theory of colour! But, before we find out how he went, let’s have a look at the mathematics behind what was known about light in Maxwell’s time. The Enlightenment – A new look at Light and Colour In 1621, Willebrord van Snel van Royen (also known as Snell or Snellius) determined that coloured light rays bend or refract when they travel between substances of different densities (like glass and air). His law, known as Snell’s Law, appears as below:

�� = ��,

Where n1 and n2 are how well the substance bends the light (known as the refractive index, with air being 1) and angles � and � are the angles at which the beam enters and leaves the substance (Figure 6a). This law helps us explain how light rays travel through lenses, microscopes and telescopes. René Descartes, applied the mathematics of Snell’s Law to investigate how rainbows form. Descartes found that when the incoming light angle is great enough, the light reflects instead of refracts, in effect bouncing back off the water droplet surface (Figure 6a, 6b). Descartes noted that light does this in a spherical water droplet once, making a primary rainbow, or twice, creating a secondary rainbow (Figure 6c, 6d). These findings were published in 1637 in his book Les Météores. a b

c d

Figure 6: a) Snell’s Law – how light is bends/refracts at the surface of water and air (Wikimedia) b) At the right angle, light reflects instead of refracts/bends. Descartes noted that light can be bent once (visible at 42 degrees) or twice (visible at 51 degrees). Forming a double rainbow (Hyperphysics) c) Sketch of double rainbows (Adapted from Descartes’ Les Météores 1637 – Plus Maths1 d) A double rainbow, consisting of a primary rainbow and secondary rainbow. Note how the secondary rainbow is considerably fainter than the primary and the colours are inversed! (Hyperphysics)

Sir Isaac Newton – Opticks and the seven colours of light In 1704, English scientist Sir Isaac Newton greatly improved what was known about colour and light through his discoveries published in his book, Opticks. Newton noticed that white light itself splits up into a rainbow of seven colours when passed through glass prisms (Figure 7). As Newton said, ‘Light itself is a heterogeneous mixture of differently refrangible rays’.11). When Newton placed these seven colours onto a spinning top he noticed that, when spun fast enough, the colours merge and white is formed. He concluded, therefore, that the reverse is also true. Combine all seven colours (violet to red), white light is made.

Figure 7: Newton proving that white light consist of multiple colours by Figure 8: Isaac Newton’s Colour circle, as produced in bending light through a prism. (Apic/Hulton Archive/Getty Images) Opticks. Notice how he has placed letters around the perimeter of the colour circle? These are to simulate each colour has a note that matches the musical scale. 10 There were multiple challenges to Newton’s Colour circle theory. Scientists and painters, such as Johann Wolfgang Goethe (Figure 9), were not happy with Newton’s explanation that all colours are made of seven colours. They knew that they could create all their paint colours using only three primary colours, surely light was the same. They didn’t like Newton’s obsessions with designating one colour to each note on the musical scale was fanciful (Johannes Kepler’s 1618 book The Harmony of the World found that nature operated in harmonic proportion, which was a very popular belief at the time. This no doubt influenced Newton to use the seven different notes of the harmonic scale). More outrage; English doctor and physicist Thomas Young, who famously discovered that light acted as light waves, found it almost impossible to believe that the human eye has a receptor for every possible colour, as this would be mean the number of receptor types would be near infinite! In 1802, Young said “it becomes necessary to suppose the number [of receptor] limited; for instance, to the three principal colours, red, yellow and blue [referring to the primary colours used by the painters]”.11 Although Young did not have quantitative evidence to support this claim, this notion set up Figure 9: Johann Wolfgang von Goethe James Maxwell Clerk, along with contemporary Hermann symmetric colour wheel. Notice how violet von Helmholtz, on the path to discovery of the true nature of and purple have been replaced with magenta10 colour.

Maxwell’s Colour Tops In 1855, Maxwell (who was working at Trinity College) decided to use this three-colour receptor theory put forward by Young. The theory states that the eye is made up of three types of colour receptors; sensitive to either red, yellow or blue light. Young’s supposed that we will be able to create any colour using different proportions of each of these primary colours. Continuing on from his mentor Forbes work, Maxwell ordered sheets of coloured paper from local Edinburgh artist D. R. Hay to test Young’s theory. His colour top (as shown in Figures 10 & 11) consisted of a spinning top with a spindle through its axis, an outer and inner disk area, where disks of paper can be overlapped and exposed as required (like a pie chart) with a measurement scale around the perimeter, to demonstrate what proportion of each colour has been used. With the colour top spinning, the outer disk, made up of a certain proportion of each colour (i.e. ultramarine (U), emerald green (EG) and vermillion (V)) would match the colour of the inner disk (for instance white). Black colour was added to the disks to remove the effects of different levels of brightness between the central disk and the outer colourful disk. If white, along with all colours in the spectrum, could be created, this was evident to confirm that Young’s three-colour theory was right.

Figure 10: Maxwell's colour top.6 Figure 11: Indicative colour layout of Maxwell’s colour top - Handprint Maxwell - Abandoning the Painter Colour Theory At Trinity College, Maxwell continued experimenting with his colour top. Placing the disks of D. H. Hay’s coloured paper on the outer disk in varying proportions to try to match the colour of the inner disk. With the top spinning fast enough, the colour top acted as an algebraic problem to be solved: find the correct proportion of each colour on the outer disk to match the colour of the inner disk. Maxwell noticed that sometimes the outer disk appeared as the same colour as the inner disk, however it appeared duller. To rectify this, Maxwell added black to make the brighter colour appear duller. He tried mixing the red, yellow and blue (what Thomas young had proposed to get the colour white), however, no matter what arrangements of each he tried, he could not reach the colour white. Somewhat befuddled, James decided to focus on only two colours instead. According to the painter’s principle colours, he knew that blue and yellow produced the colour green when mixed together. When he added blue and yellow to his disc, the colour disc, to his surprise the disk, did not produce green but a dull pink colour instead!

This was enough evidence for Maxwell to abandon the painter’s principle colours. He decided to experiment with lots of colour sheet arrangements, leading to two significant discoveries, one being that light does not act like paint and dye when added to each other (Figure ), and two, there did appear to be three special primary colours for light that when combined in the correct proportions, could produce almost any colour. Maxwell’s discovery – The primary colours of light Maxwell found that it was the ultramarine (blue), vermillion (red) and emerald green (green) that tested best during his experiments. A combination of any these discs in the correct proportions was found to create almost every colour in the colour spectrum. Maxwell had discovered the primary colours for light, we have three receptors: red, green and blue.

Figure 12: Maxwell’s colour triangle 5 R, G, B represent the primary colours red, green and blue, W represents white and r, g, b represents the components for any other colour C. In his work, he found that sheets that the colour triangle followed a simple equation: �� ℎ�� = �% �� + �% �� + �% �� where � = , � = and z= . Maxwell treated the inner and outer disk as an algebraic equation, creating white light (made of snow white (SW) and black (Bk) paper to adjust for brightness) through a mixture of vermillion (V), ultramarine (U) and emerald green (EG) colours, finding on the 6th of March 1855 in daylight that: 0.37 � + 0.27 � + 0.36 �� = 0.28 �� + 0.72 �� Maxwell could also create ultramarine (mixed with pale chrome (PC) and black to adjust for tint) using a combination vermillion and emerald green: 0.39�� + 0.21� + 0.40�� = 0.59� + 0.41�� Upon experimentation, Maxwell’s colour triangle was created (Figure 13). To use this triangle, locate the proportion of x, y and z along the x, y and z axis, where the intersection of these values is the colour that will be produced. Over the course of these experiments, Maxwell found that: 1. He agreed with Helmholtz theory that light combines in a manner different to pigments, which can be explained by Herschel’s theory of pigment absorption. 2. Confirmed that the human eye is tri-chromatic, with the combination of three primary colours in the correct proportions being able to account for most other colours in the colour spectrum. However, the primary colours are red, blue and green; not red, yellow and blue, as Thomas Young had suggested. 3. Some subjects at Cambridge who performed the colour top experiment were insensitive to the colour red. Maxwell went on to diagnose four cases of colour- blindness. Maxwell went on to publish his findings in the paper, On the Theory of Colour Vision, as perceived by the eye, with remarks on Colour-blindness 6 presenting these results to the Royal Society of London in 1860, for which he was awarded the Rumford Medal.

Figure 13: Maxwell’s colour triangle. Any colour can be made by a certain Figure 14: (a) The combination of the primary colours of proportion of red, green and blue. The green axis (z), red axis (y) and blue paints and dyes [cyan, yellow and magenta] is subtractive axis (x) - Handprint (b) The combination of the primary colours of light [red, green, blue) is additive

Teaching Resource 1: Maxwell’s Colour Tops – Manipulatives and Classroom Applications Colour Top Construction There are multiple ways that a colour top can be created. The disk templates that have been designed fit onto a compact disc (CD). Please find disk templates as attached. Equipment: • CD/ cardboard/ dip container lid • Pen or alternative for spindle • Generous blob of blue tack • Colour disks templates – printed out in colour (glossy improves experiment) • Scissors (for cutting out templates). We need to get the wheel spinning quite quickly to trick the brain into seeing one colour, a combination of the colour disks placed on the colour top. As this is the case, you can use an electric fan or drill to demonstrate this process to the class (wear glasses). Activity 1 – The Colour Wheel In this activity, student’s will develop their understanding of decimals, fractions and interpreting diagrams. 1. Start lesson off by showing (https://vimeo.com/130333096 ), OR world’s first photograph and story, optical illusion (https://www.stevespanglerscience.com/lab/experiments/american-flag-optical- illusion/ )OR story of Maxwell. 2. Students to cut out disk templates for RGB disks and small white and black disks. 3. Students to cut out 10 partition grid and then increment in decimal and fraction around the outside of the circle (Refer to examples – as attached). 4. Students to assemble colour tops in groups of two to four. 5. Show students the Maxwell Colour Triangle – question what they think it represents. 6. Students are then to create colours using the proportions dictated by the Maxwell Colour Triangle, then spin the colour top to see if their colour matches. Students to work in fractions, decimals and percentages. 7. More advanced students, calculate the area of each portion using the sector area formula.

An example spinner in motion Rationale Many scientists and mathematicians, such as Hertz, Maxwell, Einstein, Helmholtz, Herschel, Young, Snell, Newton and Descartes, were captivated by the problem of the nature of colour and light. These scientists helped us, piece by piece, to solve light’s true nature. Its properties becoming more and more like the work of science fiction, from Aristotle’s gradients of light and dark, the colour white going from something pure as to be mucked up by the accumulation of numerous strands of chromatically differing light, to Young’s wave nature of light along with the physiological trickery our eye and brain plays into the perception of light. Although an abstract concept, colour and light can be taught in class using hands on concrete and virtual manipulatives (i.e. Maxwell’s colour tops, Snell’s Law) so students can reconstruct and discover the science and mathematics behind these breakthroughs. I believe deeper learning is created when students not only learn a mathematical concept, but the historical, contextual and personal situation behind the concept too. By teaching about the history and life of James Clerk Maxwell, along with his discoveries, students too will appreciate Maxwell’s genius and have fun along the way. Australian Curriculum Links This activity can be used for Year7 students so they can work with real numbers, learning how to add, subtract, round decimal numbers percentages and fractions and being able to convert between them (ACMNA152-8; ACMNA173). Years 8 & 9 can investigate the relationships between circles, circumference and area to determine the proportion of each coloured area (ACMMG197). There is a science curriculum link, with light and energy in Year 8 (ACSSU155) and the nature of light in Year 9 (ACSSU182). The spinner can also be utilised for generating data as a primary source for chance and statistics (ACMSP169). Extension and enrichment This learning resource omits the history of light and colour in antiquity, not due to the lack of it, far from it. The author would like to mention here that there are some very interesting philosophical discussions of light made by the Egyptians and Greeks (the ‘magic mirrors’ of Egyptian tombs http://www.ancientpages.com/2011/04/21/mystery-of-ancient-magical- mirrors-some-of-the-strangest-objects-in-the-world); Aristotle holding onto the belief of the wave nature of light, analogous to sound http://photonterrace.net/en/photon/history/), and considerable progress made by the Middle Eastern scholars (Abu Ali Hasan Ibn Al-Haitham – who discovered the mathematics of refraction, and creating the world’s first pair of spectacles)12 and the impressive mirror ‘death ray’ of Archimedes (http://web.mit.edu/2.009/www/experiments/deathray/10_ArchimedesResult.html), each of which is recommended for additional exploration for learning and teaching activity. These ideas can be pursued to provide a deeper appreciation for the development of mathematics, and allow for the study of trigonometry, measurement and geometry. I would highly recommend considering incorporating a classroom experiment using the rainbow, like Descartes did back in the seventeenth century. A rainbow appears as a cone of light unique to each observer; the primary and secondary rainbow appearing at approximately 42 degrees and 51 degrees in relation to the angle of the sun. This can a simple and exciting application of trigonometry classroom exercise to confirm this angle of the rainbow. For enrichment, one can calculate why this is the angle it appears at. Unfortunately this is not as simple as calculating the critical angle at which light reflects internally off a boundary layer between a water and air: due to atmospheric water droplet’s spherical nature, there is a concentration of light that in effect produces a rainbow. The mathematics for this can be found in the University of Harvard’s “Problem of the week: Week 81, Rainbows, Solution” (https://www.physics.harvard.edu/academics/undergrad/problems). It is a delightful read that uses differentiation, geometry, and trigonometric identities in its pre3sentation of the solution to the problem. In the words of A.C. Grayling, I would suggest that you read “Rainbows” in the bath, it’s a great read and I’m sure you’ll agree, the mathematics reveals some truly surprising natural phenomena (e.g. the colours of the secondary rainbow are inversed). There is a comprehensive collection of all the different rainbows that can form in the atmosphere here: https://www.atoptics.co.uk/bows.htm ).

Additional Facts about James Clerk Maxwell A hologram of Maxwell can be found in the James Clerk Maxwell room in the Royal Society of Edinburgh. A Maxwell (Mx) is a unit measuring magnetic flux. The BBC Radio 4 podcast In our times discussed the life and ideas of James Clerk Maxwell: see http://www.bbc.co.uk/programmes/p005491g Podcast of The BBC Radio 4 podcast Science stories explains the thought experiment “Maxwell’s Demon”: see http://www.bbc.co.uk/programmes/b07dm8tb) There are three portraits of James Clerk Maxwell held in the UK : see http://www.npg.org.uk/collections/search/person/mp00914/james-clerk- maxwell?search=sas&sText=james+clerk+maxwell&OConly=true

Figure 15: A concept image of NASA’s X-57 “Maxwell” aeroplane, Figure 16: The gap between Saturn’s rings, known as Maxwell’s powered entirely by 14 electric motors and batteries. These next Gap, is currently unexplained. One popular theory is that a small generation planes will be quieter, more energy efficient and not moon may lie in this gap, however no moon has been found - NASA / reliant on fossil fuels. JPL / Public Domain https://www.youtube.com/watch?v=I0FszZoFIb0 - NASA Graphic / NASA Langley/Advanced Concepts Lab, AMA, Inc.

References 1. Descartes, R. (1637). Les Météores. (J.-M. Tremblay, Ed.) Quebec: University of Quebec. Retrieved from http://www.uqac.uquebec.ca/zone30/Classiques_des_sciences_sociales/index.html 2. James Clerk Maxwell Foundation. (2017). Facts about James Clerk Maxwell. Retrieved from James Clerk Maxwell Foundation: http://www.clerkmaxwellfoundation.org/html/key_facts_about_maxwell.html 3. Longair, S. L. (2008). Maxwell and the science of colour. Philosophical Transactions of the Royal Society A, 366, 1685-1696. 4. Lourakis. (n.d.). Plaque with Maxwell's Equations - CC BY-SA 4.0. Retrieved from https://commons.wikimedia.org/w/index.php?curid=40513486 5. Mahon, B. (2003). The Man Who Changed Everything: The Life of James Clerk Maxwell. Chichester, UK: John Wiley & Sonds Ltd. 6. Maxwell, J. C. (1855, March). Experiments on Colour, as Perceived by the Eye with Remarks on Colour-Blindness. Transactions of the Royal Society of Edinburgh, pp. 275- 298. 7. National Museums of Scotland. (2017). Disks from James Clerk Maxwell's colour top. Retrieved from National Museums of Scotland: https://www.nms.ac.uk/explore-our- collections/stories/science-and-technology/james-clerk-maxwell-inventions/james-clerk- maxwell/colour-disks/ 8. Open Culture. (2016). Behold the Very First Color Photograph (1861): Taken by Scottish Physicist (and Poet!) James Clerk Maxwell. Retrieved from Open Culture: http://www.openculture.com/2016/08/the-very-first-color-photograph-1861.html 9. Scottish Engineering Hall of Fame. (2017). James Clerk Maxwell (1831-1879), physicist whose work is the foundation of electrical engineering. Retrieved from Engineerin Hall of Fame: http://www.engineeringhalloffame.org/profile-maxwell.html 10. Tate . (2014, July). How to spin the colour wheel, by Turner, Malevich and more. Retrieved from Tate : http://www.tate.org.uk/context-comment/articles/how-to-spin-the- colour-wheel 11. Turnbull, H. W., Scott, J. F., Hall, A. R., & Tilling, L. (1959-1977). The correspondence of Isaac Newton. (Vol. 1). Cambridge, UK: Cambridge University Press. 12. Zghal, M., Bouali, H.-E., Lakhdat, Z. B., & Hamam, H. (2007). The first steps for learning optics: Ibn Sahl’s, Al Haytham’s and Young’s works on refraction as typical examples. Engineering School of Communication of Tunis.

Suggested Resources/ Reading for Teachers The first place I would start is Malcolm S Longair’s “Maxwell and the science of colour,” a non-technical review of Maxwell’s contribution, and the history of light and colour. 3 The man who changed everything: The life of James Clerk Maxwell, by B. Mahon5, is easy to read, and an in-depth analysis of James Maxwell’s life written in the context of his discoveries. To construct Maxwell’s spinning top experiment and access original papers, go to http://www.jimworthey.com/archive/index.html There are several resources you can use; however, I based my experiment design on Handprint’s resource https://www.handprint.com/HP/WCL/colortop.html Suggested Resources/ Reading for Students A brief overview of the life of Maxwell can be found at the “Foundation of James Clerk Maxwell” 2. For a virtual resource, see Maxwell’s triangle – interactive activity, which requires internet connection and java, and best opened in an Internet Explorer browser http://www.efg2.com/Lab/Graphics/Colors/MaxwellTriangle.htm

Light and Colour (J. M. W. Tuner, 1843, Light and Colour (Goethe’s Theory) – the Morning after the Deluge – Moses Writing the Book of Genesis. Source: Wikimedia Commons.)