### Why do this problem?

This problem is useful for opening students' minds to the links that exist between science and mathematics.

Note that several of these images are also found on our Scientific Measurement problems, which will allow students to measure and estimate various physical attributes of the images.

### Possible approach

This problem will very likely lead to interesting and rich discussion from the prompts in the question.

As you lead the task it is important to be aware that there is no 'right' or 'complete' answers to any of these parts. Clearly, no teacher will be an expert in all of the underlying mathematics and science, so it is very acceptable to say 'I don't know' in response to questions. You might find that some students in the class emerge as experts in various areas of interest. All these students to share their knowledge with the class. The teacher's role in this class is as a skilled facilitator: you want to help students to feel comfortable with sharing their ideas and exploring the links which emerge between mathematics and science.

To help you to guide the discussion, we have included descriptions of the images (counting across from top left) along with some points of notes. These are not in any way an exhaustive list!

### Key questions

Zebra: Why black and White? Why stripes?

Chloroplast (plant cell under magnification): What shapes form the structure? Why? (hexagons tesselate)

Honeycomb: What shapes form the structure? Why? (hexagons tesselate; compare with choloplast).

Pinecombs: These can be approximated as a cone, but also exhibit spiral structure.

Representation of a protein: Proteins are very long and narrow and fold up into tight 'blobs' to preserve space. The way proteins fold is subject to intense mathematical analysis and requires the use of very powerful computers.

Coccolith (single-celled organisms making up chalk): This shape is approximately spherical but the circles overlap in a way which is closely approximated by a dodecahedron.

Snail: Snails have shells which grow in a 3D-spiral pattern

Plant cells (from a beech tree): Similar to the chloroplast. Note the irregularities which have emerged from the straight hexagonal packing.

Diatom (An alga under magnification): This is based on a regular octagon. Note that the angles are not exactly 45 degrees; this might be because the image is a 2D snapshot of a real 3D object; each of the legs is a cell. (see http://en.wikipedia.org/wiki/Asterionella for more detailed information_)

Fin whale from the air: Note the the cross-section of this whale has an axis of symmetry and can be closely approximate by a kite.

Pollen: The pollen particle shows amazingly beautiful mathematical structure, based an a C-60 buckyball.

Vetruvian man: Leonardo Da Vinci sketched this image and points to the fact that the proportions of man are related to the Golden Ration.

Snowflake: Snowflakes grow as accumulations of ice crystals; on a very fine scale they are hexagonal in structure.

DNA (unravelled): DNA is usually tightly wound together similar to proteins. This DNA has been unravalled and we can see that it is long and very thin. As it fold up knots and complicated 'topologies' can occur.

### Possible extension

A lovely extension task is to get students to create their own posters of 16 mathematical images from natures, along with an accompanying description of the images, what mathematics underlies the images and what natural purpose the mathematical structure serves.

### Possible support

Don't worry too much about the names of the objects and why they take these shapes: look for the mathematics and symmetries in the images.