Monthly Archives: October 2016

Maths in Art?!

With the upcoming deadlines for essays fast approaching, today I found myself procrastinating… Again! Often I find myself scribbling down patterns and shapes without even thinking. Art to me is a good escape as there are no right and wrong answers, unlike maths which can seem very rigid. Today after I finished my scribbling, I realised that without my conscious knowledge at the time, throughout my drawing I had been considering numerous maths concepts.

In a recent lecture with Wendee we discussed the presence of mathematics in patterns. We discussed shapes, repetition, turns and drops. All of which seemed somewhat important in art but which artists may not necessarily deem as maths. It was clear to us that maths is extremely important in art as it allows you to calculate spacing, angles, patterns or sequences, this is something I have never considered before. To me, maths has always been a subject which was completely set aside from others.

When drawing today I found myself calculating which angle was best to hold the pen in order to achieve thinner or thicker lines. I also had to consider the spacing between the lines in the drawing itself as I wanted to keep them as parallel as possible (again, had I

Who knew there could be so much maths in a simple drawing?!

Who knew there could be so much maths in a simple drawing?!

not had a basic understanding of fundamental maths I would not understand the concept of parallel lines, or angles). The speed I moved the pen at also impacted the thickness of the lines and the accuracy of the lines which is something I had to consider more closely when the spacing of lines reduced. More maths!

 

I even found myself using maths when trying to make a random pattern. I wanted the small dotted pattern to seem random, this would imply that I did not think about how many dots I was putting in each square. However, I found myself counting the amount of dots and trying not to repeat the same number too often. I am genuinely amazed at how much mathematics can be found in even the simplest of drawings.

In a recent lecture with Anna Robb we were informed about ‘The Golden Ratio’. This is something I have never heard of until now, but on reflection used multiple times throughout my time doing standard grade then higher art in high school. The Golden Ratio is also known as Divine Proportion and is determined by the number 1.618 (Phi). The Golden Ratio is often used in art when artists are creating portraits, by distancing the facial features in relation to the golden ratio an artist considers their work to be seen as beautiful (this is often why we see artists stepping back from their work and using their brush to check proportions). A person who’s body is in proportion to the golden ratio is also seen as beautiful (unfortunately for me, I do not fall in this category). One website I came across when researching divine proportion in art describes it as (http://bit.ly/2ecAsw1)    “the most mysterious of all compositional strategies. We know that by creating images based on this rectangle our art will be more likely to appeal to the human eye, but we don’t know why.” I find it fascinating that we have an innate sense to the golden ratio and are more drawn to liking this that use divine proportion. It is even suggested that the Egyptians applied the golden ratio when building the great pyramids, as far back as 3000 B.C. Clearly, mathematics plays a fundamental role in both art and architecture which to me is extrodinary as I feel this is something we perhaps take for granted.

So far, this module has really opened up my eyes as to just how much we use maths, sometimes without knowing. I feel that because I have a firm knowledge of basic maths I don’t often think about when I am using it. Just simply being able to tell the time and calculate how long it will take to walk to a lecture means I am able to leave my flat at a suitable time. This is something I have never really considered as ‘maths’ as to me it comes as second nature. Liping Ma

Time = A basic concept we take for granted everyday

Time = A basic concept we take for granted everyday

describes having a fundamental understanding of basic mathematics as being able to identify connectedness between mathematical concepts. In my art I was able to connect the speed of the pen to the spread of ink on the paper, resulting in the thickness of the line. Moving the pen faster resulted in a thinner but less accurate line. Interconnectedness is also apparent when calculating what time to leave. You must know the distance you are travelling, the approximate speed you walk at and be able to calculate the time it will then take you to cover the distance. Although often taught together, speed distance and time are three separate yet basic concepts in maths that to be able to understand fully must be learnt together.

 

 

Maths is all around us, there is no escape!

Maths... There is no escape.

Maths… There is no escape.

Baffling Base Systems

The Base Twelve-System

I found binary much easier when adding the place value at the top of each column.

I found binary much easier when adding the place value at the top of each column.

Trying to forget what everything you have previously understood about maths is extremely challenging. This was the case when we attempted to understand the dozenal-base system. When 12 means 10 and 13 means 11 we were overwhelmed. Our understanding of maths has been built using the base ten system, which actually restricted our learning of the base 12 system. Although to us the dozenal-base system seems extremely confusing, it actually has some advantages over the base-10 decimal method of counting. 12 is a highly composite number — the smallest number with exactly four divisors: 2, 3, 4, and 6 (six if you count 1 and 12). This means that using fractions is much easier, diving into halves, thirds and quarters is much easier using twelve. (½ = 6, 1/3= 4, ¼ = 3) where as diving 10 into fractions is much more complex ( ½ =5, 1/3 = 3.33, ¼ =2.5). The dozenal system is exceptionally friendly to computer science, in fact the dozenal system is all around us. George Dvorsky explores this and says:

“a carpenter’s ruler has 12 subdivisions, grocers deal in dozens and grosses (12 dozen equals a gross), pharmacists and jewelers use the 12 ounce pound, and minters divide shillings into 12 pence. Even our timing and dating system depends on it; there are 12 months in the year, and our day is measured in 2 sets of 12.”

12 equals 10?!

12 equals 10?!

 

This course has taught me that there is so much more to maths that meets the eye. Before this, I had never explored different number systems. One benefit of the base 10 system is that it is extremely easy to count due to humans having 10 fingers and 10 toes. Zero serves as the placeholder in the base-ten system.

Binary

Another number system we have explored recently was described as the most simple number system, Binary. Although it had been described as simple, understanding the concept of base-2 system proved more than difficult to us as again our knowledge of the base 10 system restricted our learning. It wasn’t until we started writing down the system on a piece of paper that it actually began to click. This also reminded me that teacher’s should explore a topic in several ways as what works for one child won’t work for others. I think the reason that I personally struggled with this was because we were using the numerals 1 and zero which have different values in the binary and base 10 system. Perhaps if I was to ever explore this with children I would use letters to begin with. Richard spoke to us about the English schooling system and how primary aged pupils learn binary and are tested on this. The teaching of binary is important as it is the system used in computing and technologies. I found binary much easier to understand when writing it out myself with the place value written above each column.

I think it is important to explore different base systems as it can deepen our understanding of the world. Binary helps us to understand codings in computing systems, the base 60 system is used in time and the base 12 system is used in bakeries and calendar months.

When it finally clicked

When it finally clicked

Narrow-minded Maths

Our recent lecture really got me thinking about how narrow-minded we can be when considering mathematics. Before 2000px-babylonian_numerals-svgtoday’s input I had never really considered how our number system came about or in fact where maths in general came from. I found that discovering other number systems extremely interesting and creating our own was good fun too. This was also an opportunity to use some mathematical knowledge with creativity. Like many other groups in the class we tried to create a number system that symbolised the value of the number itself (1 dot for 1, 2 dots for the number 2, etc.), to us this seemed very logical. However had we placed our number system in front of someone from the Babylonian era, they would be completely baffled by our sense of logic.

After reading ‘Alex’s Adventures in Numberland’ I have discovered that the Babylonians were infact the first people to introduce a place holder. The Babylonians used a base 60 system (which is much different from our base ten system). The first column is for units, the second was for 60s, the next for 3600s. Comparing this to our system, (1, 10, 100, 1000) I feel like we have certainly developed a much more easily understandable system.ishango_bone

I have never really paid much attention or thought as to where maths originated from. Nor have I spent anytime researching ancient maths techniques and systems. After the lecture I decided to further research the Ishango Bone.

The Ishango bone was found in 1960 and is suspected to be more than 20,000 years old. Research claims that the bone is the fibula of a baboon and it was found on the shores of Lake Edward in the Ishango region. On first glance it looks like the bone is covered in tally marks, but there is clearly more to this artifact than simple tallies. Scientists believe that these marks are more than tallies, they are infact an indication that the Babylonians had a much deeper understanding of mathematics. Some scientists suggest that these marks follow a mathematical succession, others interpret the marks as some form of rule. What they can agree on is the fact that maths existed 20,000 years ago. This I find fascinating and extremely eye-opening as I’d been so narrow-minded in terms of maths believing it was something that had evolved and that all countries use a similar system.

Reference:

Bellos, A. and Riley, A. (2010) Alex’s adventures in numberland. London: Bloomsbury Publishing PLC.

Can animals really count?!

It seems that many pet owners claim that their animal can count. This is not a new idea, it seems the idea of animals counting goes back centuries. In the late 1800’s Wilhelm Von Osten came forward with the proposal that his horse, later named ‘Clever Hans’ had the innate ability to count. When Osten spoke a number, for example, the number seven, Hans was able to tap his hoof seven times, seemingly without any guidance. People were in awe of the animal’s ability to count out numbers up to ten and the horse travelled country to country showing off

Clever Hans: Image from https://upload.wikimedia.org/wikipedia/commons/5/57/Osten_und_Hans.jpg

Clever Hans: Image from https://upload.wikimedia.org/wikipedia/commons/5/57/Osten_und_Hans.jpg

his apparently extremely rare talent. However when animal experts and psychologists analysed both the horse and owner’s behaviour it soon became apparent that the horse wasn’t counting, merely just tapping until receiving some sort of signal from his owner as to when to stop tapping. Leaving the question still unanswered, can animals really count?

 

 

Clever Hans is just one example of an animal associating a word with an action. When Hans heard certain words (1-10) he knew to tap his foot, just like most dogs associate the word “sit” with the action of sitting. All of this training is done through positive reinforcement.

More recently however the question of whether animals can count or not seemed to be answered when ‘Maggie the Counting Dog’ appeared on American TV screens in 2008. To me it seems crazy that a Jack Russell can count as I can barely get my five-year-old Jack Russell to sit or roll over. Nevertheless, owner Jesse had every confidence in her seven-year-old dog’s ability to count she pits her against a class of seven-year-old children for a maths test. (As seen in the attached clip)

Like Hans the clever horse, Maggie seems to have the ability to tap out the correct answer to a number of questions. Not only can Maggie seemingly add two numbers together, she can multiply, subtract and divide! She left the class stunned as she got over ten questions correct, however, left me slightly sceptical. As with Hans, Maggie’s behaviour conveyed the idea that her owner was giving a subtle signal as to when to stop tapping.

Jesse and Maggie also featured on an episode of Oprah (an American talk show) where Jesse claims that Maggie did not need taught any maths. In fact, she claims   that Maggie was born with an innate ability to count and just needed some positive reinforcement. Like Hans, animal behaviour experts were sceptical of Maggie’s supposed counting ability so conducted several experiments to test this claim. The scientist used white sound to blur out any possible secret sound signals that Maggie was receiving, this proved that there were no sound signals so the theory that Maggie could count still seemed possible. The owner was then asked to hide her hands and covered her eyes with sunglasses, again, Maggie the counting dog answered the equation correctly. Questions were then raised when Jesse left the room and Maggie was unable to give a correct answer to several questions. It was then concluded that Jesse was infact giving Maggie some sort of clue as to when to stop tapping and although the dog couldn’t count she was deemed extremely clever with her ability to pick up subtle clues.

Several other experiments exist where scientists try to prove and disprove whether or not animals can count. Some argue it is simply a memory test for some animals, or the ability to associate and action with a word, while others argue and try to prove that animals do have an innate ability to count. it does seem that certain animals have some understanding of numbers, such as mothers counting their young, this could also be disputed as to an animals ability to smell, sense or estimate how many young surround them. Although I am slightly torn on the answer to this question it does seem that animals are extremely clever individuals and their list of talents seem to be endless. Whether or not we will ever get a conclusion to the answer remains to be seen.

Full clip of Maggie’s interview with Oprah available at:  http://www.oprah.com/oprahshow/maggie-the-dog-does-math