Monthly Archives: November 2010

The Following posts

The following posts were originally part of a website called MathsYear2000 which was created that year by the UK Department of Education.

The site was changed to http://www.Counton.org which, because of lack of funding is a shadow of its former sense.

 

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Sliceforms – the background

Sliceform models are three-dimensional objects created by slicing a solid many times in two directions. The models are a series of cross sections which are made of a set of planes cut from card slotted together.

Most Sliceforms are continuously deformable from the extremes of two flat shapes which form interesting designs in themselves. The intersections of the slices act as hinges.

The pictures you see here do not show their full beauty. You have to make them and play with their endless shapes and see the way light plays on them as you move them about and deform them. As you move the Sliceforms they change colour dramatically at times. This is apart from the constantly varying and interesting shadows they cast.

Who invented Sliceforms?
The Sliceform technique originated with a mathematician called Olaus Henrici who taught in London at the end of the nineteenth and early twentieth centuries. He made models using cross sections of quartic surfaces; these are similar to a sphere but have with cross sections which are ellipses, hyperbolae or parabolae.

Models were constructed for sale in Germany by the firm of Martin Schilling. These were designed by Alexander von Brill in Darmstadt.
The method for making the models has not been fully exploited, although it has been used for making packing for fruit and other small regular items.

In the nineteenth century mathematical models were made for teaching and understanding geometry. Many museums have collections of these.

The Strange Surfaces exhibition at the Science Museum in London contains Sliceforms from the 19th century to the present day (by Brill and John Sharp)  and other interesting historical models of surfaces. Visit their website for more details.

John Sharp has extended the system to a wide range of surfaces and polyhedra, under the name Sliceforms.

Ten minute Sliceforms

See also the post How to make and assemble the models and Sliceform templates and Downloads

The following models are simple ones for you to learn to assemble the models.
In many cases there are more detailed versions or variations which take longer to cut out and assemble because they have more slices.

These models take about ten minutes each to make.

Do not think that because they take such a short time to make that they are any less interesting or complicated to make.

Try them first before you make the more advanced models.

Possible explorations
Use them to explore variations with different coloured sets of slices.
Can you make right and left handed versions by colouring?
What types of symmetry do they have?

The models

More advanced Sliceforms

See also  the post on how to make and assemble the models and Sliceform templates and Downloads

The following models are either more advanced versions of the Ten minute Sliceforms or ones with many slices comparable to the ones in John Sharp’s Sliceform book.

These models take about an hour to each to cut out and assemble, depending on how experienced you are.

Even if you have made Sliceforms before, try the Ten minute Sliceforms before you make the more advanced models. In most cases having the simple version to hand will help you when you come to make the more advanced ones.

The models

Books and Poster on Sliceforms

John Sharp has two books on Sliceforms published by Tarquin.

Sliceforms


Sliceforms is a set of eight models to cut out and make.
The models in the book have similar numbers of slices to the advanced models described here.  There is a short description of creation of models.

Surfaces: Explorations with Sliceforms

Surfaces is a more advanced book. It describes various techniques for creating Sliceforms from purely artistic methods to mathematical ways of modelling surfaces using this technique.

Sliceforms Poster

The Sliceforms Poster is available as thick paper and also in a laminated version.

CLICK ON THE LINKS ABOVE TO GO TO THE TARQUIN SITE TO ORDER THEM

 

How to make and assemble the models

Please read the following instructions carefully, especially the way to cut the slots. If you just cut slits, with a single cut instead of a slot as described below, then the model will buckle and will not deform properly. The following results compare well cut slots on the left with the effect of only cutting slits on the right.

How the templates are provided

Templates for each model are provided as files to print. You can either print directly onto card or print and then photocopy.

Files are provided as Adobe Acrobat PDFs

The reason for supplying files like this, rather than presenting them as web pages is to get good quality fine lines. To show them as good enough quality would mean large files that take a long time to download. Also the line thickness has been adjusted so that, when you cut out the slices, you leave the minimum construction line on the model.

This way you can have files ready to print at any time, or prepare masters for classroom teaching.

Printing or copying onto card

The templates are for printing or copying onto photocopy card (160 gsm).
The page size of the model is A4 or US letter size. Make sure you select the correct template for the page you are using.
Many simple models can fit onto to a single page so templates are provided for making many models from the same sheet of card, for example for classroom teaching, or a sheet is provided with more than one model.

The larger models with more slices are designed to print on two sheets of card. In some cases the slices are the same on both sheets, whereas for others the two sheets are different. Please read the instructions for each model.

With the larger models, use different colours of card for the two sheets. By making copies on two cards and using one colour in each direction, you will get much better models; they are also easier to make.

Do not try to enlarge the models, they will not work as well.

Cutting out the slices and cutting the slots

When you cut out the slices, cut just inside the line so there is no drawn line visible on the final model.

In most cases it is best to cut out one or two slices at a time, then cut the slots and then assemble them. Some templates have numbered tabs to show you the order of the slice. Cut these off when the model as been assembled.

It is very important that you cut slots and not just slits. You must cut just enough for the card making the slice to fit neatly in the slot. If the slot is too much wider, the model is likely to fall apart easily, especially in the early stages of assembly. If you just cut a slit along the line, then the forces acting sideways on the slices will cause the model to buckle. Slots look like this:

Cut the slot with scissors making a pair of cuts either side of the line. This will give a sliver of card which will often curl up like a hair. Pinch it off at the end of the slot by grasping it with your fingernails.

To practice cutting slots, use some spare card from the edge of the template with a short pencil line drawn on them.

A Sliceform tetrahedron

See also the posts on how to make and assemble the models and Sliceform templates and Downloads

This Sliceform is available in two versions, a simple one which takes about 10 minutes to cut out and assemble since it has only six slices and a more advanced one which has 18 slices and takes about an hour to make.

You should make the 10 minute one before attempting the advanced one.

How the model is designed

This model has been designed by taking a tetrahedron in a cube like this:

and making two sets of slices, one horizontal and one vertical, parallel to sides of the cube.

These slices are all rectangles. See at the end of the post  for why. It should help you to assemble the tetrahedron Sliceform.

The 10 minute tetrahedron template

Download the file you want for printing and use that to create the model. The following graphic is to help you understand what the slices look like.

The slices for half of the model look like this:

A single page of the template contains slices to make four tetrahedra. Make two copies, each on different coloured card and mix the different coloured slices for different effects, to give eight variations on the tetrahedron. Start by making all slices in each direction a different colour.
See How to make and assemble the models for general printing instructions for printing or copying onto card.

Assembling the 10 minute tetrahedron

See the post on how to make and assemble the models for more about cutting out and cutting the slots. Take extra care with the slots.

The simple model only has six slices. The two central squares that correspond to the squares in the design model and two other slices in each direction.

Since there are only a few slices, cut out all the pieces and cut the slots on each one.

Fit the centre squares together, then add the other two slices in each direction. Use the two half tetrahedron models to judge how they are orientated. Think symmetrically.

When you have made the model, fold it flat in two directions.

The advanced tetrahedron template

Download the file you want for printing and use that to create the model. The following graphic is to help you understand what the slices look like.

The slices for half of the model look like this:

A single page of the template contains slices to make one tetrahedra. Make two copies, each on different coloured card and use one colour in each direction to make two models.

See the post on how to make and assemble the models and Sliceform templates and Downloads for the general instructions for printing or copying onto card.

Assembling the advanced tetrahedron model

This model has many more pieces and takes up to an hour to make. With more slices the tetrahedron looks more solid.

Make the simple model first so that you can see the structure when you are making this one.

Take extra care not to cut the slots too wide or you will find that the pieces fall out in the early stages of assembly. You will probably find it easier to cut pairs of slices and then fit them onto the model.

Cut out the central squares and their slots and fit them together. Then cut the next four slices (two in each direction) and add them to your model. Continue cutting and adding four slices at a time until you reach the outside. If you make the model with two colours of card it will not only be a more interesting model, it will also be easier to make.

When you have made the model, fold it flat in two directions. With it folded, tap the long flat ends gently on a flat surface to distribute the slices evenly in the slots. Look at the patterns it creates.

The slices of a tetrahedron and why they are rectangles

This is a picture of a conventional net for half a tetrahedron.

Cut two versions out on paper and assemble them by sticking the tabs.

Placing the squares of the two halves together makes a tetrahedron (if you put them together with a 90 degree rotation).

The dotted lines show the position of slices for making the Sliceform.
Measure them and trace them round to see how the shapes of the rectangles change, from a long thin one at the edge, to a fatter one as the sides become equal in the central square. As you continue to the other half, they become longer and thinner again, but in the other direction.

It is a good idea to make this model to help you see how the rectangles of the Sliceform slot together.