Boy’s Surface was discovered in 1901 by German mathematician Werner Boy. Like the Klein Bottle, this object is a single-sided surface with no edges. Boy’s Surface is also a non-orientable surface, which means that a two-dimensional creature can travel within the surface and find paths that will reverse the creature’s handedness when it returns to its starting point. The Möbius Strip and Klein Bottle also have non-orientable surfaces.
Formally speaking, Boy’s surface is an immersion of a projective plane in three-dimensional space with no singularities (pinch points). Geometric recipes exist for its creation, and some of them involve the stretching of a disk and the gluing of the disk to the edge of a Möbius strip.
In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having a mean curvature of zero. The term “minimal surface” is used because these surfaces originally arose as surfaces that minimized total surface area subject to some constraint. Physical models of area-minimizing minimal surfaces can be made by dipping a wire frame into a soap solution, forming a soap film, which is a minimal surface whose boundary is the wire frame. However the term is used for more general surfaces that may self-intersect or do not have constraints.
Art by Paul Nylander.
Geometric art by Mark A. Reynolds
Reynolds on his work:
There is a sympathetic magic between pencil and paper that is primordial. Many people have had an experience with this magic in some way. Bring geometry to the magical ceremony that we call drawing, and the power and energy of the union can be sublime and infinite, especially when geometric structures are drawn. The shapes, forms, grids, and structures that I sometimes call compositions are almost endless. Through decades of work with geometry, I have come to realize that drawing is a transformative act, and, that drawing geometrically is also a transcendental one.
Erwin Panofsky, in his Idea, a Concept in Art Theory, said that Plato conceived of the Idea as being, “in the world of shapes and figures something perfect and sublime, to which imagined form those objects not accessible to sensory perception can be related by way of imitation”. These shapes and figures of Plato’s were essentially geometric in form. If we consider the geometric aspect of reality and the structures of the universe, we are led to ponder just when geometry first became present in this universe, and that geometry may indeed be a gift from the gods. One may be led to ponder whether geometry, by its very nature, within its chi, there is a consciousness similar to our own. At least, there may be an awareness that geometry itself is a vehicle that connects the human mind with the universe of things and energies. Its mechanisms, its ways of structuring, of composing, may be inherent within geometry itself, for we did not invent geometry, we discovered it. (Perhaps there was an exception on mankind’s obsessive use of the straight line however.) I believe that geometry at least shares something with the human mind that makes mind aware of the eternal, the constant. We then ask when geometry first came into existence in the universe, and by what hand or energy or law.
Different modes of oscillation for a pendulum
The period of a simple pendulum is not a trivial thing, and it depends on the initial conditions.
Shown here are ten different modes of oscillation for the same pendulum. The only difference is the total amount of mechanical energy in the system.
As a result, each one has a completely different period of oscillation, unlike what the small-angle approximation (as taught in high-school) would suggest. They can’t be in sync. You may see some really interesting patterns based on the delay between them in your browser.
The red graph above each pendulum represents the phase portrait for the respective mode of oscillation, with the current state marked as a blue dot. The horizontal axis represents angle (hence why it wraps around the sides) while the vertical axis represents angular velocity.
Pendulums are very interesting dynamical systems, as they are relatively simple to understand but can produce surprisingly complex results in certain cases, such as the chaotic behavior of double pendulums and the odd behavior displayed by coupled pendulums.