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Spider Webs
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Spider Webs
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Back to Home Page Back to Spiders back to Nature’s Maths Here are two pictures of threads from garden spider webs taken through a microscope.
The first one shows a cylindrical thread with lemon shaped blobs. A long piece of web thread will usually carry a large number of these blobs, spaced and sized so uniformly that the arrangement looks perfect. The second picture shows that a thread can also carry a series of alternately larger and smaller blobs. Again the spider usually produces an apparently perfect thread. This picture was chosen because it shows two consecutive large blobs, which spoil the sequence. Here is a picture showing the junction of a radial thread with a part of the spiral thread. The radial threads support the spiral which carries the blobs.
These blobs are actually the sticky part of the web. They are made of a viscous liquid, which gathers itself into the lemon shape because that is the shape of miminum energy. The shape of the outline of a blob is a mathematical curve called an unduloid. The family of unduloids includes shapes ranging from very thin to almost spherical, depending on the diameter of the thread and the volume of liquid in the blob. In practice the blobs usually look similar to these examples, presumably because there is some advantage to the spider. The shape of the curve is a result of the equality of pressure throughout the blob, which means that the total curvature at all points on the surface must be the same. This does not mean that the blob must be a sphere. The total curvature is the sum of the curvatures in two planes at right angles, so these can vary individually. On a damp morning, drops of water may form on the blobs. Water Sometimes there is one drop per blob: at other times the drops may hang on two blobs each. The weight of the water makes the threads droop into graceful shapes. If the blobs are equally spaced and the drops are of equal weight, the curves of the threads are close to being catenaries. A catenary is the shape of a cable before a bridge is hung on it, or the curve of an empty clothes line, in the absence of wind. Catenaries are actually related to unduloids. If the deck of a suspension bridge is vastly heavier than the cables, the curve is almost a parabola, which is a simple mathematical curve. In practice the cables have significant weight, and so the shape is intermediate between a catenary and a parabola. Here is a picture of the M4 suspension bridge across the river Severn. See also Bridges and Funicular.
The cross-sections of the threads and blobs of a web are circles, which are also simple mathematical curves. So spider webs appear to be very mathematical in several respects – they can include catenaries, circles and unduloids. In fact they are not mathematical at all; the shapes arise automatically because of simple physical constraints, without any specific action by the spider, deliberate or otherwise. In this respect the webs differ from the honeycombs of bees, in which the cells are arranged to use smallest possible amount of wax while retaining enough strength. The bees make hexagonal cells. There is no physics to make this happen automatically – the bees are programmed to do it. To obtain a piece of spider web for examination through a microscope it can be captured on a glassless 35 mm slide frame. The first picture in the set above shows a small section of a support thread. It carries water drops. They are much more spherical than the sticky blobs, because they are much larger. But their outlines are also close to being unduloids. At this scale, the tiniest particle of dust is enough to distort the drops from the perfect shape. In the case of the blobs, they were formed as the thread left the spider’s body, before any foreign objects had time to land on the thread.
The shapes above are the result of a calculation of some of the curves in the unduloid family. The ones near the middle represent the shapes of the sticky blobs, while the larger ones represent the shapes of water drops on a thread without blobs. If water drops hang on a thread with blobs they can be much larger, because the blob provides a much larger area for them to cling to, as compared with a bare thread. In that case the drops are close to spherical, like the outer curves in the diagram above. The outlines of the blobs and drops are not actually complete unduloids. An unduloid is composed of many of these segments joined end to end, forming an undulating curve. In the physical example, the more sharply curved the shape, the higher the pressure difference across the interface. If the pressure difference is zero, the curve is a catenary. Starting at the bottom of the curve and going out, it never curves over and comes down – it diverges for ever and does not oscillate. This type of curve can be made using two coaxial circular wires to collect a film of soapy water. The process of changing the parameters of the curves from unduloid to catenary can be taken further. Beyond the catenary the curvature becomes stronger, and the curve bends over and forms a loop. Instead of a wave-like shape, the curve, called a nodoid, forms a series of loops. There are some beautiful photographs of water drops on spider webs in the book "Closeups in Nature" by John Shaw. This book is full of sound advice and wonderful photographs of many kinds of plants and animals. Both the spider web and the bridge, like many other structures, are mainly empty space, because the optimum distribution of material is hardly ever as a lump. Here is the result of all that engineering – a trussed fly. What makes the sticky material gather itself into little blobs? And what makes water sometimes form little drops and sometimes spread all over a surface and wet it? Surface Tension Water Spider web photographs Australian spiders Spider
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