MATHS AND ART Mrs Michelle Hobbs

MATHS AND ART

Mrs Michelle Hobbs

“The mathematical sciences particularly exhibit order symmetry … these are the greatest forms of the beautiful.” Aristotle

Many great artists have used mathematical ideas and concepts either consciously or subconsciously. An awareness and, better still, understanding of the mathematics involved can add another layer of appreciation and enjoyment when studying a painting or a building. There are a multitude of ways that Mathematics can be interwoven with Art. Sometimes formal mathematical rules and concepts can be applied such as the Golden Ratio, Perspective and Tessellation, but often a more general appreciation of shape and pattern is present. The intersection of Art and Mathematics can also be seen in Fractal Art. Stunning images can be produced when numbers generated by an iterative mathematical process are identified with particular colours. This article will focus on Geometry and Art by way of introduction to a much wider field.

To begin, consider the basic mathematical shapes: circles, triangles, squares and rectangles. Artists like Mondrian and Kandinsky combined basic shapes with colour to striking effect.

Mondrian ‘longed for an art of clarity and discipline that somehow reflected the objective laws of the universe’.

Kandinsky used geometric forms ‘in his quest to achieve harmony and to express a spiritual reality’. For him coloured shapes could be identified with feelings and in Yellow-Red-Blue it is clear to see how he juxtaposes light and dark shapes.

Kandinsky and Mondrian both used the simplicity and clarity of geometry in their search for order and harmony and I feel that their paintings sit very happily in a Mathematics classroom. This tradition of considering abstract form and the influence of Mathematics can also be seen in the sculptures of Barbara Hepworth. Visiting her garden in St Ives is an excellent opportunity for appreciating the fusion of Mathematics and Art. It really is a multisensory and multidisciplinary experience.

Modern western art is not the only art to rely heavily on geometry in its search for beauty. Islamic artists have long been influenced by Mathematics and Science in their use of geometric shapes to convey unity and order. Tiling in the Nasir-ol-Molk Mosque in Shiraz, Iran, illustrates the use of circles, squares, stars and polygons. These have been tessellated, rotated, reflected, enlarged and translated to create intricate, complex patterns. Something to bear in mind when revising IGCSE transformations!

An article on Mathematics and Art would not be complete without mentioning the fraught topic of the Golden Ratio. The Golden Ratio was observed by the Ancient Greek Mathematician, Euclid, and artists such as the polymath Leonardo Da Vinci may have used it in the Last Supper and other paintings.

However, the evidence for the use of the Golden Ratio knowingly being linked to what is aesthetically pleasing is not always clear-cut. Its apparent use could often be unconscious or coincidental, but it is still quite fun to explore as Dali realised in his painting, The Sacrament of The Last Supper. There are many websites that claim that the use of the Golden Ratio will enhance your art and appreciation of art history, but I will focus on explaining some of the mathematics involved.

The Golden Ratio is the ratio 1: Φ (Phi) where Φ is the solution to the quadratic equation: x2-x-1=0. The first few digits are 1.618… but the digits go on forever as it is an irrational number. There is also an interesting link to the Fibonacci sequence as the division of successive terms tends to Φ. A rectangle whose sides are in the ratio 1: Φ is called a golden rectangle and removing the largest square contained in the rectangle leaves a smaller rectangle, which also has sides that are in the Golden Ratio. This subdivision results in the Golden Spiral as explained here by Alex Bellos: ‘We can go on for as long as we like subdividing rectangles. And if we draw quarter circles in each square, we get a spiral. The illustration below is probably one of the most famous images in mathematics, if not in all of science. The curve is called the “golden spiral”. (Strictly speaking the golden spiral is a smoothed-out version of this curve) …’

Overlaying this spiral on paintings like Da Vinci’s Mona Lisa and Hokusai’s Great Wave could suggest the use of the Golden Ratio and Spiral; key features of the paintings do seem to fit spirals and rectangles.

It is even claimed that the Greeks had the Golden Ratio in mind when constructing the Parthenon in 447 BCE. An internet search will yield interesting results or come and look at the poster of Hokusai’s Great Wave in classroom M3.

Needless to say, I believe that Mathematics underpins most of our lives. Artists and architects have long understood its beauty and the interplay between Mathematics and Art can enhance our appreciation of both.

IMMERSE PRIZE

An external competition with Immerse Education, where participants have the opportunity to showcase their subject knowledge and essay writing skills, with the chance to win a full scholarship to one of their 2022 summer programmes in Oxford, Cambridge and London.

IMMERSE PRIZE: WHY DO WE NEED CYBER SECURITY?

Maria Taraban, UIV

Cyber security is the protection of computerised data, networks and technological devices from potential threats, such as illegal utilisation of technology or malicious interference (known as cyber-attacks). Some forms of cyber security include, for example, cloud security offered by Microsoft OneDrive, application security to control access with a strong password, and operational security, which includes contingency planning for cyberattacks. Cyber security is vital in protecting businesses, essential establishments and individuals from cyber-attacks.

For businesses, cyber-attacks can be destructive in a number of ways. Firstly, cyber-attacks put companies’ reputations in jeopardy when the news on the security violation becomes public. This can lead to consumers doubting the security procedures of the companies, causing a decline in purchases. Dangerously, cyber-attacks can mean precious data becomes inaccessible. For example, Code Spaces, a code-hosting company, encountered a malware that deleted a lot of valuable data. As a result, it had to return money to customers, which made it financially unstable, bringing about the company’s closure. According to Experian, a consumer credit reporting

company, around 60% of small enterprises shut down following cyber-attacks. Cyber-attacks are therefore responsible for unemployment rates increasing. Additionally, closure of companies means suffering of the overall economy of a country, as consumer rates decrease. Decreased consumer spending leads to weak aggregate demand and a falling growth rate. Consequently, cyber security is required to decrease the possibility of unemployment and to stabilise a country’s economy.

Cyber-attacks could severely disrupt functioning of establishments that serve society, such as hospitals, police, fire departments and other essential institutions. The cyber-attacks can cause problems, such as financial loss from theft of money and inaccessible IT systems. As an example, Springhill Medical Centre in Alabama experienced a cyberattack and could not access its IT systems for over three weeks. This affected staff communication and made maternal heart monitors unclear. The ambiguous heart monitors affected at least one new-born, as the infant died nine months after birth from ‘subsequent brain damage’. Generally, cyberattacks make vital institutions less accessible for people in need, increasing the risks of death. Hence, we need cyber security to reduce death rates from cyber-attacks.

Finally, a type of cyber-attack called ‘phishing’ can disadvantage individual people. Phishing could be implemented by offenders trying to obtain someone’s personal information by targeting potential victims through emails, while impersonating a genuine organisation. These emails often deceive people into disclosing personal information, such as bank details, by offering money and valuables in return. According to the FBI, in the USA, there were 241,342 registered phishing occurrences in 2020. Phishing scams are detrimental because they can end in theft of identity or money. To keep safe from these menaces, cyber security is needed.

To conclude, cyber security is crucially important, as it lowers the risk of theft or loss of valuable data, guards business operations from paralysis, protects IT systems of societal establishments and makes people less vulnerable to scams. Most industries and public sectors require a degree of cyber security to ensure general safety.

IMMERSE PRIZE: WHAT IS HISTORY’S MOST SIGNIFICANT INVENTION?

Cheuk-Yi Cherie (Sage) Lau, UIV

REFERENCES

Encyclopedia.com. “Magnifying Glass.” Medical Discoveries. 2018. [https:// www.encyclopedia. com/science-andtechnology/technology/ technology-terms-andconcepts/magnifyingglass last accessed: 5 Jan. 2022].

History.com. “8. Magnifying Lenses” in “11 Innovations That Changed History”. Andrews, E., 2021. [https://www. history.com/news/11innovations-thatchanged-history last accessed: 5 Jan. 2022].

Classroom.synonym. com. “Who Invented the Magnifying Glass”. Stover, E. [https://classroom. synonym.com/ invented-magnifyingglass-5031039.html last accessed: 5 Jan 2022].

Brown.edu. “Historical Context of the Magnifying Glass.” In “13 Things”. 2008. [https://www.brown. edu/Departments/ Joukowsky_Institute/ courses/13things/7334. html last accessed: 5 Jan. 2022].

Sciencing.com. “How Do Magnifying Glasses Work”. Crystal, M., 2017. [https://sciencing.com/ characteristics-planemirrors-7220163.html last accessed: 5 Jan. 2022].

Nasa.gov. “What Is the Hubble Space Telescope”. Wild, F., 2021. [https://www. nasa.gov/audience/ forstudents/k-4/stories/ nasa-knows/whatis-the-hubble-spacetelecope-k4.html last accessed: 5 Jan. 2022] Human beings are innovative. We love to develop and manufacture new articles, constantly flooded with the latest prototypes and items on the market to buy to make our lives easier. Amongst all of these lies the magnifying lens and, although it may not seem the most remarkable invention at first, it is easy to underestimate this product and its contributions to society.

The Ancient Egyptians were pioneers, breaking up fragments of crystals and obsidian to view smaller objects. In addition, Roman Emperor Nero used gemstones to view performances in the amphitheatre. The Roman philosopher Seneca also described his use of water in a bowl as a magnifying tool. However, it was not until the thirteenth century when the first lenses came into use. Roger Bacon, an English philosopher, wrote about and experimented with the magnification of glass spheres to assist the weak-sighted scholars who needed to continue pursuing their teaching and research at the University of Oxford. This led him to create a magnifying glass which inspired and assisted the creation of the first spectacles in 1280 Florence. However, this is still debated as the original creator is still not found. By the sixteenth century, holding a glass magnifier to produce an enlarged image was widespread and this guided the invention of optical instruments. Telescopes and microscopes were introduced in the late sixteenth century, a major innovation that enhanced the fields of biology and astronomy. The invention of the microscope facilitated Robert Hooke, the first ever to observe micro-organisms such as cells, and Antonie van Leeuwenhoek, a pioneer and often regarded as the ‘father of microbiology’. A magnifying lens is made up of a convex lens. Light rays enter the lens in parallel but refract and bundle all parallel rays to the focal point. This changes the path and enlarges the angle of incidence of the rays, appearing larger. It does not move closer or further to the eye and is only stretched in our perspectives.

The introduction of magnifying lenses has also inspired other ingenious products to be created, such as the electron microscope, a powerful tool to inspect biological specimens, and the Hubble Space Telescope, designed by NASA to explore the universe by taking detailed pictures, allowing us to observe the formation of planets in our Solar System with much more clarity and understanding.

On a much smaller scale, loupes are comprised of three magnifying lenses and are devices that assist jewellers to see gems and other precious stones to evaluate their value and help photographers to see how clear an image is when magnified. This proves that magnifying lenses can be beneficial in a vast number of fields.

The magnifying lens is often used as the backbone of other inventions. It is a tool that assisted the creation of other products that we consider essential in all fields. From photography to optometry, it has played a major part in shaping our past and we should never doubt its significance.

IMMERSE PRIZE: THE HISTORY OF THE REFRIGERATOR

Pearl (Ivie) Avwenagha, UIV

According to the Cambridge English Dictionary, an invention is ‘something that has never been made before, or the process of creating something that has never been made before’. Before the eighteenth century, there have been many ground-breaking inventions to do with preserving food for the public, such as the action of canning, salting, potting and freezing. However, out of all the methods and inventions created at that time period, I think the refrigerator was the most significant.

B. Krasner-Khait, ‘The Impact of Refrigeration’, History Magazine, [http://www.historymagazine.com/refrig. html]

J. Bandy, “How did the invention of the refrigerator have an impact on history? How did it make life easier, and what other things did it lead to?” eNotes Editorial, 14 March 2010. [https://www.enotes. com/homework-help/ how-did-inventionrefrigerator-have-animpact-147519 last accessed 31 Apr. 2021.]

The History of the Refrigerator, M. Bellis, 01 November 2019, [https://www. thoughtco.com/historyof-refrigerator-andfreezers-4072564]

The History of the Refrigerator, M. Bellis, 01 November 2019, ThoughtCo. [https:// www.thoughtco.com/ history-of-refrigeratorand-freezers-4072564] In modern life, the refrigerator is a component used constantly every single day and, for most people, it would be very hard to imagine life without it. Prior to the 1830s, salting, spicing, drying, pickling or smoking were some of the only ways to go about food preservation. Refrigeration was not needed, because the foods it preserved, for example fruits and vegetables, were not an important part of the North American diet, as they are currently. There were also iceboxes – food was left in a box lined with tin or zinc and an ice brick was placed on top to melt and cool the box. However, iceboxes’ success entirely depended on how fast the brick of ice melted. Very often, it would melt too fast. According to The History of the Refrigerator, ‘A refrigerator produces cool temperatures by rapidly vaporising a liquid through compression’. These household appliances have a thermally insulated compartment which transfers heat from the inside of the refrigerator to the outside, making the inside cooler.

In 1748, Scottish scientist William Cullen demonstrated the first form of refrigeration at the University of Glasgow. Despite his ingenious invention, it was never used for any practical purpose. In the mid-1800s, an American inventor named Oliver Evans designed a blueprint for the first refrigeration machine, but it wasn’t built until 1834, by Jacob Perkins. Using a vapor compression cycle, the refrigerator created cool temperatures to store food in. Ten years after this, John Gorrie, an American physician, based a new refrigerator off Evans’ design. He cured yellow fever patients, using the air from the device. Despite all of this, we owe the process of liquefying gas, that is part of basic refrigeration technology today, to German engineer Carl von Linden, who patented it in 1876.

In conclusion, I believe the most significant invention in history is the refrigerator, because many people had a more healthy and well-balanced diet than before as the refrigerators could preserve the food for longer. There were also many medical problems related to the use of natural ice: the original water was polluted. In the future, I think the current model of refrigerators could be improved even more, as the refrigerants used are very potent greenhouse gases. Maybe a model could be developed using solar power, or other renewable energy.

IMMERSE PRIZE: THE CENTRAL SCIENCE

Anya Gannon, LVI

At our most basic, we are all just a collection of chemicals reacting to produce specific responses.

Chemistry is the most important scientific field partly due to its historic significance, and its undeniable symbolism as mankind’s thirst for curiosity and knowledge being most profoundly fulfilled. However, it is not the content of Chemistry that proves its overarching importance, but the skills acquired when learning it.

Biology at a high level can mostly be explained through a chemical perspective such as the unique properties of water and the use of medicinal drugs in the body. Although biological principles and procedures are not taught in Chemistry, the transferrable skills of problem-solving and piecing together various forms of information mean that a chemist can help a biologist with ease. Physics is a more abstract science that explains everything due to principles and therefore, it lacks importance due to the constraints of these principles and their inefficiency to be showcased clearly at a real-world level. Indeed, density equals mass over volume; Chemistry allows us to understand why ice is less dense than water which the other two sciences may struggle to decipher.

Another reason the principles of Physics cannot be reasonably compared to Chemistry is that they create a tunnel vision where there is nearly always a correct answer and correct method. Chemistry includes taking many variables into account and developing reasoning skills to explain the nature of chemical reactions or the failures of experiments, thanks to the deep understanding that there are many factors that can influence outcomes. Chemistry can be viewed as superior to Physics as it takes the principles and can utilise them effectively to discover the capabilities of these invisible rules. For example, pressure is a force that participates greatly in the formation and structure of nearly everything around us in a wholly passive role, yet Chemistry appreciates this power and unleashes its potential in the hydration of ethene whereby pressure is used as a key component to produce ethanol which is essential in a variety of industries. Chemistry also plays a vital role in the future of sustainability and creating ways for us to adapt to the environmental crisis that is currently being experienced. The use of biofuels in Brazil as an energy source is an example of a way that Chemistry leads to a path of increasing the efficiency of our current resources and decreasing their negative impact on the world. Chemistry provides us with awe and bewildering appreciation for the world around us by both simplifying and complicating every structure and movement. There is a chemical explanation for almost every situation including those not designated to the scientific field, including the formation of addictions and emotions such as love through the release of dopamine and its chemical interactions in the body.

Nevertheless, the greatest feat of Chemistry that highlights its importance is its ability not to stand alone in ignorance of all other sciences but to utilise them to explain everyday marvels. It is the central discipline that lends skills and borrows them, to unite a plethora of principles and experiments to make sense of a world full of nonsense.