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Fermat’s Principle, Reflection, and Refraction
Published in Vasudevan Lakshminarayanan, Hassen Ghalila, Ahmed Ammar, L. Srinivasa Varadharajan, Understanding Optics with Python, 2018
Vasudevan Lakshminarayanan, Hassen Ghalila, Ahmed Ammar, L. Srinivasa Varadharajan
The study of propagation of light in geometric optics includes the concept of rays and wavefronts. A ray, a mathematical abstraction, is considered to be a line perpendicular to the wavefront, and can essentially be thought of as a thin pencil of light. A ray can be thought of as a propagation of energy in the limit of wavelength going to zero. Since the wavelength of visible light is of the order of a few hundred nanometers and the dimensions of optical components such as mirrors, prisms, and lenses are of the order of millimeters or centimeters, we can neglect the finiteness of the wavelength. Under such conditions, we enter the realm of geometric optics. Rays (and wavefronts) can either be divergent (as from a point light source), convergent (as when light is focused on a point), or parallel. Figure 5.1 illustrates these ideas. We can state unequivocally that the duty of any optical element (be it a simple lens, the human eye, or the Hubble space telescope) is to change the vergence of the wavefront, or equivalently, the direction of the incident light rays.
Geometric problem solving with strings and pins
Published in Spatial Cognition & Computation, 2019
Christian Freksa, Thomas Barkowsky, Zoe Falomir, Jasper van de Ven
From a cognitive science perspective, it may be interesting to note that although constructive geometry appears more tangible than other fields of mathematics, it takes place entirely in the conceptual domain of mathematics. ‘Compass’ and ‘straightedge’ correspond to abstract notions connected with abstractly defined properties. The two notions merely serve as metaphors to support our imagination. This becomes apparent when we note that the tools for geometric constructions do not mention paper, pencil, or other media through which to apply the construction; accordingly, neither the compass nor the straightedge need to leave a trace in form of a pencil line. For geometric reasoning, it is sufficient to know that intersections between lines can be constructed and therefore exist and that we can use them as concepts for mental constructions. Paper and pencil serve to support our imagination and reasoning processes through generating a perceivable analogy of the conceptual structures and operations and to help us to communicate geometric concepts to fellow human beings.