Position and Other Geometries Ch. 5, page 6

Position and Other Geometries

Other Geometries

When we are dealing with 2D and 3D shapes we are generally working within a geometric system called Euclidean geometry (named after the famous ancient Greek geometer Euclid). In this geometry, lengths and angles are important and remain unchanged (invariant) with actions operating on them. Lengths and angles are important in identifying the shape. There are, however, other geometries and it is worth realizing that in those actions may not be creating a change. The following diagram illustrate this. This is not really part of the school curriculum but it helps us to place our geometric knowledge into a bigger picture. You will see some overlaps. For example, isometry illustrates that reflections or slides or turns do not change anything other than position. Similarity (enlargement) changes size but not angles. The final one, topology, changes each aspect except neighbourliness.

Invariant under mapping / Position / Length / Angle and ratio / Parallel
ism / Cross-ratio / Neighbour-liness
identity / · / · / · / · / · / ·
isometry / · / · / · / · / ·
similarity / · / · / · / ·
affinity / · / · / ·
perspectivity / · / ·
topology / ·

Figure 1. Geometries and building diagrams.

Topology in Societies

Topology in Primary Schools

Topology is a geometry in which directions and size do not stay the same under change. In topology it is the connection and betweenness that is important. For example, imagine a drawing on a sheet of rubber which is pulled and stretched and distorted. The angles and directions change but the eye is always inside the head circle, the two lips still join, and so on. A cup could be distorted in three-dimensions to form a donut - afterall both have only one hole.

This idea of order in connections is used in networks. A common example of a network map is that of the train lines in a city. The order of stations and connecting stations are correct but the distances and directions are not too accurate. To make it easy for passengers they can be roughly in the right direction but in a true network map this may not be the case.

The diagrams for assembling parts of a computer arrangement, a VCR and TV etc, are similar types of maps.

Network theory is important, for example, in transport designs and in electronics. Unless the network is only traceable one way, it will not work for electric currents.

Below are a range of network diagrams.

Figure 1. Networks.

The famous puzzle of the bridges of Koisenberg was established by the famous mathematician Euler. The people of the city liked to promenade of an evening. The question was could they cross all the bridges in the river without retracing their steps and arrive back at their starting point.

Here is a diagrammatic view of the river, its bridges, and islands.

Figure 2. Network of the bridges of Koisenberg.

In the last syllabus revision, topology and knots became an optional topic. Knots have many practical experiences. Sailors, truck drivers, Scouts, hairdressers, bush sports, and crafts like macramé all use knots. These should be covered in any case whether or not the simpler mathematical understandings are followed up. Unfortunately, after a few simple experiences the study of topology becomes quite difficult.

Learning Tasks for Readers

Topology Activities

·  Which diagrams above are traceable without lifting your pen.
·  Which ones are the same topologically?
·  Are the paths from the riverbanks, islands and bridges traceable?
To help decide on when they are traceable or not, you need to check how many points have an even number of sprouting lines (called even vertices) and how many an odd number. If you keep a record of the number of odd and the number of even vertices in a table, you can work out an important element of network theory; that is when are networks traceable.

Aboriginal Maps

Interestingly, these kinds of maps were used by Aboriginal people for many thousands of years. The important aspects were the main waterholes and campsites with connecting lines. The arrangements were often symbolic to the people and used in special ways in art.

Figure 3. Schematic Aboriginal art.

Continuous curves

Sand drawings are common in many cultures including Australian Aboriginal groups. Other designs come from Vanuatu (Ascher, 1991). Often the curves are a short curve repeated in a rotational or other symmetrical pattern like the one below. Continuous curve designs are common in Scandinavia, Central Europe, Africa, and the Americas. They can be found on quilting.

Learning Tasks for Readers

Repeated Design Activities Creating Space and Using Position

·  Try copying the design below. The basic design is copied in vertical and horizontal reflections.
·  Make up a similar right-angled basic
motif and try reflections.
·  On Drape you can draw this basic pattern, then turn and repeat to see what happens.
The design from Vanuatu is a sand drawing with a repeated motif, reflected and turned. Can you draw it?
Figure 4. Vanuatu design (Ascher, 1991).
·  Continuous curves are common in Europe and for quilting. This one comes from a First Nation group in America. Can you trace the design?
Figure5. American First Nation design.

Cats Cradles

These string games are found in Africa and among Australian Aboriginal groups. They have patterns build on symmetrical and asymmetrical actions with the fingers dropping or collecting strings on both or one hand. Different results occur. Here are some examples.

Figure 6. Australian Aboriginal string designs.

Other traditional examples can be found in Mozambique with a paper by Marcos Cherinda that can be found on http://www.uog.ac.pg/glec/index.htm under scanned papers held by GLEC.

Knots

Knots are another part of topology. If a piece of string is placed on the table it might be a true knot or a false knot. This is illustrated in Figure 7. Students need to be able to follow diagrams of knots as these are used in many jobs and recreational activities from sailing to macramé. Most students do not have trouble watching a person make a series of knots and follow them just as they can learn to knit or crotchet. Friendship bands are one popular knotting activity. Below are some Scandinavian knots.

If the ribbon is pulled tight, will it make a knot?

Figure 7. Diagrams of knots.

Figure 8. Scandinavian knots.

Pacific Islands Bands

The Maori tapire is made by folding and weaving as illustrated below. It makes a pleasant armband. To do this you may follow someone who shows you how to do it and you will need to follow this 2D representations of 3D actions. This is an important skill. The two pieces are woven at 60o. This is illustrated in chapter 3 on angles.

Origami

Japanese people pass on to their children the joys and skills of origami. Many designs are used to portray a relationship, hope, or feeling. Chapter 2 gives one example for a house and there are many books illustrating how to fold paper to make Australian animals, cat’s head etc.

Creating Space: Professional Knowledge and Spatial Activities for Teaching Mathematics Kay Owens