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Topology refers to the study of properties of shapes and spaces that are the same topological properties when stretched, bent or twisted. This particular area of mathematics has applications in even the simplest parts of simple life. From the simplicity of a child tying shoelaces to the complexity of quantum physics, topology is prevalent everywhere. Since its introduction in the 1800s, the development and understanding by mathematicians has progressed significantly. As this development has taken place, the concept of continuous deformation has come along with a plethora of other concepts.
Figure 1: Formation of a Donut from a Coffee Mug
The Concept Of Continuous Deformation
By definition, continuous deformation is the gradual change in the shape of an object without changing its basic structure and topological properties. Let’s look at 2 historical examples which have revolutionised topology.
Figure 2: The Möbius Strip, a ring of paper with a half-twist
The Möbius strip revolutionised the topology space due to its simple assembly and the fascinating one-sidedness of this topological object
Now, What relevance does the Möbius strip have towards continuous deformation?
Well, the true answer is in the exploration of the fact that it only has one side. If the strip were to be stretched, compressed or twisted, the strip would retain this property. The strip cannot turn into a regular ring without being cut.
Figure 3: Torus, a tube-shaped object
-The Torus is formed by rotating a circle around a 3-dimensional plane. It revolutionised topology by linking it to geometry which gave mathematicians a wider understanding of the niche topic at the time
- How does it link to continuous deformation?
Similar to the Möbius strip, the Torus has a property which cannot be changed if compressed, stretched or twisted. However, instead of being one-sided, the Torus’ property is that it has 1 hole. This property stays the same, even if the object is deformed.
Conclusion
Our exploration in this article consisted of Topology and 2 examples which reinforced the idea of continuous deformation. The simplicity of our examples gave us an outlook into topology, however, as this topic is explored deeper, the examples get much tougher to comprehend and are still challenging the top mathematicians to this day.
Citations:
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University of WaterLoo (2015). What is Topology? [online] Pure Mathematics. Available at: https://uwaterloo.ca/pure-mathematics/about-pure-math/what-is-pure-math/what-is-topology.
What is Continuous Deformation (2018). What is Continuous Deformation? [online] Mathematics Stack Exchange. Available at: https://math.stackexchange.com/questions/2679338/what-is-continuous-deformation [Accessed 30 Oct. 2024].
Wikipedia Contributors (2019). Möbius strip. [online] Wikipedia. Available at: https://en.wikipedia.org/wiki/M%C3%B6bius_strip.
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