It was decided that we would skip the more conventional Orchard Road as we felt that many of the other groups would be using examples from there. Therefore we decided to go down to the Raffles City / Suntec City area to search for images of mathematical interest.
The two pictures above show a tiling pattern we found at Marina Square. We saw two points of mathematical interest in the tiling. Firstly, one can see the octagon formed by the eight bordering rectangular tiles, as can be seen from the picture on the left. However, what we found more interesting was that the unit in the left picture was part of a basic unit in the tiling of the floor in the formation of an interesting tessalating pattern, as can be seen in the picture on the right.
Even though we had not covered symmetry and patterns in the course, we decided to check out the resources available on the course website. We found out that this type of symmetry is known mathematically as p4 symmetry, which means that the pattern consists of a non-centred cell with 4-fold rotational symmetry and no reflectional or other symmetries.
The photograph on the right is a sculpture we saw at Marina Square. What is interesting about this sculpture is that it exhibits different planes of reflectional and rotational symmetry in 3D. In the abovementioned photograph, one can clearly see one of the planes of reflection of the sculpture. One could also deduce from this photograph, albeit wrongly, that there may not be other points or planes of symmetry in the sculpture. The reason why we say this is that we were also "tricked" at first glance of the sculpture. However, as the three bottom photographs show, there are more points and planes of symmetry in the sculpture. The leftmost photograph in the bottom row of photographs shows another plane of reflectional symmetry of the sculpture. This other plane of symmetry was noticed accidentally as we circled the sculpture in hopes of finding other interesting mathematical concepts besides the aforementioned. |
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The middle and the rightmost photographs show two of the points of rotational symmetry of the sculpture. However, if one examines the issue carefully, these two points of rotational symmetry actually exist on the same point in space. This is because as can be seen in the photographs, the sculpture is rotated about its centre to achieve the two "different" rotational symmetries. The misconception of there being more points of rotational symmetry could arise from the fact that the sculpture is 3D in nature, but we tend to analyse symmetry in 2D planes.
The next two photographs show the back and front windows of the concierge area at The Ritz Carlton Millenia, Singapore. The photograph on the right is of the main entrance of the hotel itself. One can see from these photographs a six-sided shape made up of rectangles, triangles, trapeziums and hexagons inscribed in a circle.
At first we thought that the pattern may be a simplification of the (34, 6) or (3, 4, 6, 4) Archimedian tilings, but a quick check confirmed that it could not be so. This is because when one sub-divides the rectangles into squares and the trapeziums into equilaterial triangles in order to form regular polygons, one can see that there are two types of vertices being formed, which goes against one of the rules of an Archimedian tiling, which states that all vertices of an Archimedian tiling must be of the same type.
We also found many references of the octagon in The Ritz Carlton Millenia, Singapore, as well as in The Fullerton, Singapore. This may be because the Chinese believe that the octagonal shape represents the Ba Gua which enhances the beneficial energy and subdues the negative energy of a location. Two of the examples are the pillars on the outside of the hotel and the bathroom windows of the guestrooms. Of special interest were the windows as most people would have thought of them as round windows at first glance.
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We found the octagon at another unexpected place; as an actual architectural form on the Millenia Tower. In this case, one cannot easily determine the shape by counting the number of sides as may be done in the previous examples. However, by moving about slightly around the block, one may observe that at certain points, only three of the faces of the extruded polygon may be seen; however, at other points four faces may be seen. From this one may deduce that the shape is indeed an octagon. |  |
The above photographs show the Suntec City logo. It is clear from the photograph on the left that the 2D version of the logo has four axes of reflectional symmetry, and also has rotational symmetry of order 4. When expressed in mathematical notation, such a symmetry of a finite design is also known as of the order D4.
What is also interesting is that this order is also preserved in the 3D version of the logo mounted on the exterior wall of the Suntec City Exhibition Hall. In fact, this order is extended to the tiling behind the logo. We see on the right photograph that the logo lies on 6x6 square tiles, and the surrounding tiles are of the same size as well. This lining-up of the logo with the tiles on the wall creates the extended symmetry which many people may not notice.
If the previous examples deal with symmetry, either apparent or not so apparent, then the Esplanade - Theatres on the Bay has many designs of apparent symmetry which under closer scrutiny are not symmetrical at all. One example of such asymmetry is the logo of the Explanade itself, as see in the photograph on the right. One easily replies that the axis of symmetry is 45° right of the normal when given a quick glance at the logo and asked to state the axis of symmetry. However, upon a more detailed look, it is seen that there is actually no symmetry in the figure; the deliberate tilt of the logo makes this detection more difficult. If the logo were arranged straight up on its supposed "axis of symmetry", it would easily be seen that there is no symmetry. One may easily test this theory out by tilting one's head at the prescribed angle before looking at the logo. |
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Another place in the Esplanade where there seems to be, but lacks actual symmetry is the half-pyrimadal glass forms as seen in the photographs above. At first glance, they seem to suggest regular glass pyramids that are bisected into halves, as in the photograph on the left. However, the large triangles formed by the small triangular and rhombic units are actually not regular; in fact, they slant towards the wall side. One is reminded of the pyramidal forms of the Musee de Lourve in Paris, France. However, unlike the forms mentioned here, the rhombic and triangular units forming the glass pyramid of the Musee de Lourve are regular.
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The photograph on the right shows a lamp at the Singapore Art Museum. The outer case is in the form of a hexagon that is slightly squashed in at the edges. Nonenetherless, the outer case by itself exhibits D6 symmetry. On the other hand, the inner lighting unit exhibits D4 symmetry. However, when the two units come together, the order of the symmetry becomes C2. This means that the object exhibits no reflectional symmetry and rotational symmetry of order 2. |
The last set of photographs are taken at the National Library. The railing pattern on the windows of the library are a stretched version of a Archimedian tiling. The tiling in question is the (4, 82) tiling. It may be observed that while the four-sided shapes in the window railings remain as squares, the eight-sided figures in the railing deviate from the actual Archimedian tiling and become stretched in the vertical direction.
