Mathematical similarity
Mathematical similarity is a foundational concept primarily explored within geometry, where objects are deemed "similar" based on shared features such as shape and proportionality. This concept transcends mere visual resemblance, asserting that mathematically similar objects maintain consistent ratios between their corresponding dimensions. The idea of similarity has been instrumental in various fields, including architecture, physics, and mathematics, offering simplified models for complex phenomena, like comparing the heart rates of different species. Historically, figures like Thales of Miletus utilized similar triangles for practical measurements, while other mathematicians like Euclid and Apollonius contributed to the formal understanding of similarity in geometric figures.
Furthermore, the study of similarity extends into modern applications, from computer graphics to mechanical engineering, where it aids in scaling and transformations. Educationally, the concept fosters critical thinking and logical reasoning in students, emphasizing the importance of proportional relationships in problem-solving. However, it can also present challenges, as students may confuse mathematical similarity with broader notions of sameness. Overall, mathematical similarity serves as a vital tool in both theoretical and practical contexts, underscoring its significance in understanding the relationships among various geometrical and physical entities.
Mathematical similarity
SUMMARY: The concept of mathematical similarity has been studied since antiquity.
The concept of “similarity” is universal, playing a particularly large role in the field of geometry. In general, objects may be called “similar” if they share features that look alike, such as shape, color, or value. However, it is a much stronger statement to say that two objects are “mathematically similar.” Similarity can be a powerful simplifying assumption in modeling situations. Scaling an object appears in many applications, such as in architecture. Scaling notions can also explain the speed of a hummingbird’s heartbeat as compared to a human heart, and why certain insects would collapse under their own weight if they were scaled to a large size. Julian Huxley asserted that the evolutionary struggle to maintain similar surface-to-volume relationships is important in anatomy. Recognizing a similar object is also important. Logician and philosopher Willard Van Orman Quine felt that learning, knowledge, and thought all require similarity so that humans can order objects into categories with similar meaning. Similarity is often connected to triangles in mathematics, starting in grades three through five, but there are many other mathematical situations where it is also useful, such as in the definition of trigonometric functions, in axiomatic arguments, in matrices, in analysis of differential equations, and in fractals.
Early History
Distance calculations contributed to the development of similarity. Thales of Miletus is said to have measured the height of a pyramid using its shadow, but historians are unsure of the method that he used. A method that makes use of similar triangles is attributed to Thales by Plutarch of Chaeronea. In classrooms in the twentieth and twenty-first century, similar experiments are conducted. By measuring the length of the shadow of a tall object, like a pyramid, tree, or building, at the same time as measuring the length of a shadow of a known meter or other stick, a proportion with similar right triangles can be formed. The method assumes that light rays are parallel. In ancient China, instruments such as the L-shaped set-square or gnomon also needed similar triangles. In chapter nine of the Nine Chapters on the Mathematical Art, problems were posed and solved using similarity concepts. One of the problems has been translated as
There is a square town of unknown dimensions. There is a gate in the middle of each side. Twenty paces outside the North Gate is a tree. If one leaves the town by the South Gate, walks 14 paces due South, then walks due West for 1775 paces, the tree will just come into view. What are the dimensions of the town?
Many other mathematicians have worked on a variety of similarity concepts and applications. In Euclid of Alexandria’s Elements, the various definitions of similarity depend on the figure being examined. Apollonius of Perga explored the similarity of conic sections. During the seventeenth and eighteenth centuries in China, the proportionality of corresponding sides of similar triangles in the plane was quite useful in solving problems in spherical trigonometry. In some twenty-first-century college classrooms, students explore the reason why spherical triangles with shortest distance paths and the same angles must be congruent—there is no concept of similarity on a sphere. Mathematics educators also study the conceptual difficulties in teaching and learning similarity.
Other concepts of similarity arose from mechanics concerns. In his work on the equilibrium of the plane, Archimedes of Alexandria postulated that plane figures that are similar must have similarly placed centers of gravity. Galileo Galilei tried to generalize the notion of geometric similarity to mechanics. Isaac Newton, Hermann von Helmholtz, Joseph Fourier, James Froude, Osborne Reynolds, Lord Rayleigh (John Strutt), and others also worked on variations of similarity in physical situations. Building on their work, and motivated by the lack of a theoretical foundation for flight research, Edgar Buckingham articulated a formal basis for mechanical similarity in 1914. Aside from physical applications, in computer graphics, transformations that preserve similarity can be used to scale mechanical and dynamical behavior in addition to static images.
In 2021, research conducted at the University of Cambridge suggested the concept of similarity merges proportional reasoning and geometry. For students, this offered a context to develop the understanding of forming a mathematical argument such as a proof. It also helped these individuals move from visual to logical reasoning. In addition, early intuitions about similarity were likely developed as children played with models and distorted 2D and 3D shapes. Conversely, a young student’s understanding of geometric similarity could be made confusing when conflated with non-mathematical discussions on concepts such as sameness.
Bibliography
Fried, Michael. “Similarity and Equality in Greek Mathematics.” For the Learning of Mathematics, vol. 29, no. 1, 2009.
Sterrett, Susan. Wittgenstein Flies A Kite: A Story of Models of Wings and Models of the World. Pi Press, 2005.
"Similarity." Socratica, 2024, learn.socratica.com/en/topic/mathematics/euclidean-geometry/similarity. Accessed 25 Oct. 2024.
"Understanding Why Similarity Works." Better Explained, betterexplained.com/articles/understanding-why-similarity-works. Accessed 25 Oct. 2024.
“What Does Research Suggest about the Teaching and Learning of Similarity?” University of Cambridge Mathematics, no. 39, Nov. 2021. www.cambridgemaths.org/Images/espresso‗39‗teaching‗and‗learning‗of‗similarity.pdf. Accessed 25 Oct. 2024.
"What Is Geometric Similarity?" Interactive Mathematics, 2024, www.intmath.com/functions-and-graphs/what-is-geometric-similarity.php. Accessed 25 Oct. 2024.