This unit examines the use of reflective, rotational, and translational symmetry in the design of logos. Logos are designs associated with a particular trade name or company and usually involve symmetry to make them aesthetically pleasing as well as functional.
- find all the lines of reflection symmetry in a given shape
- identify the order of rotational symmetry of a given shape (how many times it "maps" onto itself in a full turn)
- create designs which have reflection symmetry, rotational symmetry (orders 2, 3, 4, 6) and translational symmetry
This unit looks into the symmetry of a number of objects. The focus is on reflective and rotational symmetry, although there is some reference to translation symmetry. The concept of symmetry is fundamental to mathematics and is used extensively in various guises. Symmetry appears frequently in nature, for example, it can be seen in honeycomb, where it is based around the regular hexagon. Its importance for bees is that it gives strength to the overall structure. It is also seen in crystals that were originally classified by their symmetric structure. Again, the purpose of the symmetry in crystals is to provide strength.
Mathematics investigates these natural structures and this has led to classification of crystals, wallpaper friezes, wallpaper patterns and the like. Symmetry is frequently used to make mathematical arguments simpler. For example, if we wanted to analyse the game Noughts and Crosses, we would start by looking at the original position of the board. By symmetry, there are essentially only three squares: a corner square, a square in the middle of a side and the centre square. All corner squares are equivalent by rotational symmetry. The same can be said for squares in the middle of a side.
So symmetry is an important concept that goes through all of mathematics and its study at this level provides a basis for work that is to come.
design equipment (rulers, compasses, protractors, set squares)
paper circles (easily cut out of newsprint in batches by stapling the centre of a traced circle)
Connected 2, 1999, Samoan Siapo Pattern
- Fold a circle in half and cut pieces out of it in any way you like. Ask the students to anticipate the pattern that the "opened up shape" might have. Write down any mathematical vocabulary that might arise, in particular terms like symmetry, reflection, turn, etc. Open up the cut circle to confirm the students’ predictions. The shape has reflective or line symmetry along the fold line.
- Ask why reflective symmetry is sometimes referred to as mirror symmetry and flip symmetry. Each alternative term can be demonstrated. Holding a mirror along the fold line enables students to see an image of the whole shape even when one half is masked. Similarly, the amended (cut) circle can be refolded in half and traced around. Then turn the paper over the fold line and trace around again. The traced figure will be that of the whole amended circle.
- Students should then make their own shape by folding a circle in half and cutting out pieces (they must not cut all the way along the fold line). This will enable them to come to the generalisation that all such shapes have at least one line of reflective symmetry (some may have two depending on the cuts). These shapes may be displayed on a chart.
- Move onto folding a circle into quarters (in half then in half again), and cutting pieces out. Ask what symmetry they anticipate the new cut circle will have. Have them create their own pattern in this way then compare it with a partner to find out what symmetry both shapes have in common. Most students will realise that there are two lines of reflective symmetry (the fold lines) but many will miss the half turn rotational symmetry. This is referred to as order 2 as the shape maps onto itself twice in a full turn.
- Get the students to anticipate then investigate what symmetry the cut circle will have if folded in eighths or sixths before cutting. Sixths can be created by folding the circle in half and then looping the half into thirds (see diagram below). (The circle folded into eighths will have at least four lines of reflection symmetry and rotational symmetry of order 4; with sixths the symmetry is six lines and order 3). Ask the students if they can see a pattern in the "least number" of lines and orders.
- Challenge the students to use paper circles to create a shape that has rotational symmetry of order 3 but no lines of reflective symmetry. Next ask them to produce a shape that has rotational symmetry of order 4 but with no lines of reflective symmetry. Then ask for one that has rotational symmetry of order 5 with no lines of reflective symmetry, etc.
- Car Logos
Investigate the logos found on motor vehicles by looking at cars in the school car park, magazine advertisements or images from the internet. Car advertisements in magazines can also be cut out and used. Most manufacturers use symmetry of some kind in designing their logos. For example, Audi uses four intersecting circles in a line. This pattern has one line of reflection symmetry. This logo is created by translating one circle three times.
Discuss the symmetry of each logo found and compile a list of companies for future reference. (You may need to omit the manufacturer’s name from some of the logos in order to get any symmetry. For instance, removing ‘Ford’ from its logo gives an elliptical shape that has two lines of symmetry.)
- Construct Car Logos
Provide the students with drawing instruments such as rulers, protractors, drawing compasses and tell them to recreate the car logos they saw in the car park, online or in magazines. Have the images available for them to refer to. At times it may be necessary to bring the students together in order to discuss construction skills. For example, for a logo involving rotational symmetry of order 3, a protractor will be useful. Since there are 360° in a full turn, one third of that is 120°, which gives the angle measured at the centre for dividing a circle in thirds. Construction skills like drawing a right angle by using a protractor or compass construction may be modelled if necessary.
- Logos in the media
As homework (see Homelink) ask the students to find other examples of logos. Obviously not all logos have symmetry. Sporting goods manufacturers are good examples of this. Nike use a tick, Adidas use three stripes etc. This will illustrate to the students that logos have to be both aesthetically pleasing (i.e. often symmetric) and suggestive of the nature of the company. Share the logos they bring along and group them by symmetry discussing what message is suggested by the logo image. This has strong links to visual language in the English curriculum. For example, the Canterbury Clothing Company has a logo of three translating, overlapping C’s that resemble a ball. The three images give an impression of a single ball moving from left to right.
- Creating Logos
Set up the following scenario for the students: You work for an advertising company as a logo designer. There are four new companies for whom you have to create logos. They have stipulated that the logo must have some kind of symmetry but must also suggest what goods and services they provide. (If you wish, they may also be required to come up with a slogan that captures the message, e.g. "just do it".)
Here are the companies:
Sweeties - a company that make sugar-free lollies that taste great and don’t ruin your teeth.
Gadgets - who make neat construction gadgets (gears, blocks, wheels, etc.), so you can create your own toys.
Dudes - makers of cool clothes especially for primary school children.
Brainbuilders - the people who you see for extra help with schoolwork. You get one-on-one help so you are in a class of your own!
Give the students sufficient time to design logos for one or more of the companies. They will need to present the logo in a short report to each company that shows what symmetries are involved and how the design suggests what kind of goods or services they provide. You may decide to set up a voting system for the class to decide on a winning logo for each company.
- Enlarge the logos on the copymaster. Display them and ask the students to each write down what symmetry each design has (this is useful for assessment purposes). Get them to share what they have written in pairs then bring the class together for a collective discussion.
- For each logo get students to demonstrate what symmetry the design has by using scissors or folding and flipping, rotating. Make a list of symmetries for each design.
- Tell the students to look at their first list and add any information they may have missed. If they do this in a new colour you will have evidence of their initial independent understanding and their new shared understanding