[MUSIC] So now we've learned quite a bit about point symmetry and plane symmetry. And I think it's time that we got out of the studio and had a look at these symmetry operations in action. And in particular, it's very nice to have a look at the use of symmetry in nature. For this we're going to visit the experts at the Singapore Botanic Gardens who can tell us about symmetry in plants and also symmetry in the design of the garden as a whole. To help us understand this, we're talking today to Dr. Nigel Taylor, who is the Director of the Singapore Botanic Gardens and can explain to us really wonderfully how symmetry and botany go together. So Nigel, thank you very much for your time today. Why don't you tell us a little bit about the history of the gardens here. >> Well, Tim, we're standing in front of the main gates, the Tangling Gate of the Gardens, which itself is a nicely symmetrical object, the pair of gates, but made up of rather asymmetrical plants. In this particular case a Bauhinia which is very popular for its orange flowers. And strangely, planted around the gate here, it refuses to flower as well as it does elsewhere. You know what plants are like. They're just like animals. They never do exactly what you want. The gardens inside, however, were laid out in the 1860s by a Scotsman who was familiar with a landscape design which is based on asymmetry, informality, no two halves matching the other. And this we can see when we go inside. So we start with a symmetrical pair of gates and we go inside to a garden which is rather asymmetric in most of its parts, not all, but most. So this is Swan Lake, Singapore's oldest ornamental water body, made in 1866 by Niven, the grand designer of the Botanic Gardens. And it follows the contours. It was previously a freshwater swamp, and what Niven did, he just used the topography of the place and created this lake by damming its outflow. And then asymmetric, off center in the lake, is this island which was planted up with these palms in 1891. And so here we have a typical part of Niven's botanic garden with its asymmetry which is very pleasing to the eye in terms of imitating nature. And now we're gonna move on to the one part of Singapore Botanic Gardens which is formal and regular in shape and very symmetrical, in complete contrast to what we see here. This is the Sundial Garden begun in 1929. Various of its features were added later, but essentially it's a perfectly symmetrical structure with the sundial at its center. It's out of kilter with the rest of SBG which, as we've said, is an asymmetrical garden in English landscape movement style. But this garden also has its own asymmetry, because at each corner we have a Grecian lady, and each lady is a little different from the next. The rest, however, is as near as symmetrical as it could be, allowing for the fact that not every plant is identically the same. But we can reflect on whether the person who designed this garden, director Eric Holttum, did so consciously as contrast with the rest of the landscape or whether, indeed, he didn't realize that he was constructing something out of sequence with the rest of the asymmetrical landscape. So here we have a cactus in the Botanic Gardens, and as with all nearly all plants, plant parts are arranged in spirals. But at first sight it's rather hard to detect the spirals here because something else has happened. The spirals are these little spine clusters. They're called areoles. But that's not as important as the fact that this is where the plant's leaves would be. And leaves, as on most stems, are arranged in a spiral. What's happened here, however, is that because this plant is adapted to living in dry places which are very sunny, it's organized the elements of the spiral that are vertically one above the other on a rib. So the spiral is still there but what you see is actually a rib. And this rib has two functions. As the plant dries out, the rib can shrink and then when it gets more water, it can expand. So it's a bit like a concertina, a reservoir of water which can expand and contract without damaging the plant. But more importantly, it's also providing a self-shading device. Because as the sun moves around through the horizon, each rib shades the next one in turn. So no one part of the plant is exposed to the light for too long. On the other hand, every part of the plant is exposed to some light, so each part of the plant can indulge in photosynthesis. So it's a very clever but simple way of regulating the plant's intake of light and also with the light the amount of desiccation that the plant might suffer. So the spine clusters, organized in spirals but then superimposed on ribs, give us an interesting conundrum. Because this plant has some stems which are perfectly symmetrical, with an even number of ribs, and other stems with an odd number of ribs which are bilaterally symmetrical. In other words, you can only cut the stem in one place and get two equal halves. But this doesn't matter to the plant. The functionality is the same whether it has an odd number or an even number of ribs. What matters is that the rib functions in the ecology of the plant, shading and allowing water storage. The Singapore Botanic Gardens' most famous quality is the National Orchid Garden within it. And behind me is Burkill Hall, the 1868 black and white plantation-style bungalow which was Niven's home. And it now sits in the middle of the National Orchid Garden and is the center where we look at the VIP orchids and other special orchids like Singapore's national flower. And thinking about orchids from the point of view of symmetry, we can detect immediately that this flower is bilaterally symmetrical. It has two identical halves, but there's only that one plane in which you can cut it which gives you those two identical halves. And we may ponder on why the vast majority of orchids are structured in this way, and I think it has a lot to do with their pollinator. Orchids have to attract pollinators to ensure that the pollen from one plant is transferred to the female parts of another plant. And it's done in a very special way because orchids produce vast numbers of seeds, which means those ovules that become the seeds have to receive an equivalent amount of pollen. And the way the orchid does that is to have the pollen concentrated into a mass containing millions of pollen grains. And this mass has to be stuck to the insect in a particular way, because it's a lock-and-key mechanism, to ensure that the insect, when goes from the first flower with the pollen to the next flower to pollinate it, deposits that dob of pollen in exactly the right place. Now the insect, of course, is itself bilaterally symmetrical. And the flower has evolved this structure in order to line up the insect rather like a plane lines up when coming in to land at the airport for the runway. It has to line it up so it goes into the flower in exactly the right position in order that the head or the neck of the insect or another part will receive the pollen which can then be transferred to the next flower. And so the bilateral symmetry of this flower is absolutely essential for the pollination mechanism. This is the Dendrobium hybrid named after Princess Diana. It's a rather sad story because she passed away before she could receive it. However, we were able to show her son William, Prince William, last year this orchid, and he had a quiet moment with the orchid when he had visited the Botanic Gardens with his wife Kate. So here we're looking at a Mokara Zhu Rongji, and this is an interesting climbing orchid that produces aerial roots along the stems. And the flowers, of course, are, like all other orchids, bilaterally symmetrical and have this curious bronze pink color which is really quite unusual.