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Reflection symmetry can be generalized to other isometries of m -dimensional space which are involutions , such as. Antipodal symmetry is an alternative name for a point reflection symmetry through the origin. Such a "reflection" preserves orientation if and only if k is an even number. That is why in physics the term P- symmetry is used for both point reflection and mirror symmetry P stands for parity. As a point reflection in three dimensions changes a left-handed coordinate system into a right-handed coordinate system , symmetry under a point reflection is also called a left-right symmetry.

Rotational symmetry is symmetry with respect to some or all rotations in m -dimensional Euclidean space. Rotations are direct isometries ; i.

This does not apply for objects because it makes space homogeneous, but it may apply for physical laws. For symmetry with respect to rotations about a point we can take that point as origin. For chiral objects it is the same as the full symmetry group. Laws of physics are SO 3 -invariant if they do not distinguish different directions in space. Because of Noether's theorem , rotational symmetry of a physical system is equivalent to the angular momentum conservation law. If the line of triangles extended to infinity in both directions, they would have a discrete translational symmetry; any translation that mapped one triangle onto another would leave the whole line unchanged.

The symmetry group comprising glide reflections and associated translations is the frieze group p11g and is isomorphic with the infinite cyclic group Z. In 3D, a rotary reflection , rotoreflection or improper rotation is a rotation about an axis combined with reflection in a plane perpendicular to that axis.

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In 3D geometry and higher, a screw axis or rotary translation is a combination of a rotation and a translation along the rotation axis. Helical symmetry is the kind of symmetry seen in such everyday objects as springs , Slinky toys, drill bits , and augers. The concept of helical symmetry can be visualized as the tracing in three-dimensional space that results from rotating an object at a constant angular speed while simultaneously translating at a constant linear speed along its axis of rotation.

At any one point in time, these two motions combine to give a coiling angle that helps define the properties of the traced helix. Three main classes of helical symmetry can be distinguished based on the interplay of the angle of coiling and translation symmetries along the axis:. In 4D, a double rotation symmetry can be generated as the composite of two orthogonal rotations. A wider definition of geometric symmetry allows operations from a larger group than the Euclidean group of isometries. Examples of larger geometric symmetry groups are:. In Felix Klein 's Erlangen program , each possible group of symmetries defines a geometry in which objects that are related by a member of the symmetry group are considered to be equivalent.

Scale symmetry means that if an object is expanded or reduced in size, the new object has the same properties as the original. Similarly, if a soft wax candle were enlarged to the size of a tall tree, it would immediately collapse under its own weight. A more subtle form of scale symmetry is demonstrated by fractals. A coast is an example of a naturally occurring fractal, since it retains similar-appearing complexity at every level from the view of a satellite to a microscopic examination of how the water laps up against individual grains of sand.

The branching of trees, which enables small twigs to stand in for full trees in dioramas , is another example. Because fractals can generate the appearance of patterns in nature , they have a beauty and familiarity not typically seen with mathematically generated functions.

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Fractals have also found a place in computer-generated movie effects , where their ability to create complex curves with fractal symmetries results in more realistic virtual worlds. With every geometry, Felix Klein associated an underlying group of symmetries. The hierarchy of geometries is thus mathematically represented as a hierarchy of these groups , and hierarchy of their invariants. For example, lengths, angles and areas are preserved with respect to the Euclidean group of symmetries, while only the incidence structure and the cross-ratio are preserved under the most general projective transformations.

A concept of parallelism , which is preserved in affine geometry , is not meaningful in projective geometry. Then, by abstracting the underlying groups of symmetries from the geometries, the relationships between them can be re-established at the group level.

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Since the group of affine geometry is a subgroup of the group of projective geometry, any notion invariant in projective geometry is a priori meaningful in affine geometry; but not the other way round. If you add required symmetries, you have a more powerful theory but fewer concepts and theorems which will be deeper and more general.

Graphing Circles and Writing Equations of Circles In Standard Form - Conic Sections

William Thurston introduced a similar version of symmetries in geometry. A model geometry is a simply connected smooth manifold X together with a transitive action of a Lie group G on X with compact stabilizers. The Lie group can be thought of as the group of symmetries of the geometry. A model geometry is called maximal if G is maximal among groups acting smoothly and transitively on X with compact stabilizers, i.

Sometimes this condition is included in the definition of a model geometry. If a given manifold admits a geometric structure, then it admits one whose model is maximal. A 3-dimensional model geometry X is relevant to the geometrization conjecture if it is maximal and if there is at least one compact manifold with a geometric structure modelled on X. Thurston classified the 8 model geometries satisfying these conditions; they are listed below and are sometimes called Thurston geometries.

There are also uncountably many model geometries without compact quotients. From Wikipedia, the free encyclopedia. Main article: Reflectional symmetry.

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Main article: Point reflection. Main article: Rotational symmetry. Main article: Translational symmetry.

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Main article: Glide reflection. Main article: improper rotation. See also: point groups in three dimensions. See also: Screw axis. Transformation Geometry: An Introduction to Symmetry. Strategic communication ensures that the impact ofyour message is consistent with your intentions, and results in understanding. What you say, the way you say it, where, when, and under what circumstances it is said shape the performance culture.

When leaders maximize their contribution to daily conversations, they engage and align people around a common cause, reduce uncertainty, keep people focused, equip people for moments of truth that create an on-the-table culture, prevent excuses, learn from experience, treat mistakes as intellectual capital, and leverage the power of leadership decisions to shape beliefs and behaviors. Leaders translate vision and strategic direction into goals and objectives, actions and accountabilities. Performance accountability systems clarify what is expected of people and align consequences or rewards with actual performance.

Leaders need to build discipline into their leadership process and management cycle to achieve accountability, predictability, learning, renewal, and sustainability. The best organizations develop simple processes that are internally efficient, locally responsive, and globally adaptable. Complexity is removed from the customer experience to enable them to engage you in ways that are both elegant and satisfying. Establishing and optimizing operational performance is an ongoing journey.

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Operations need to be focused on the priority work, using the most effective techniques—aligning initiatives and operations with strategy; continuously improving operations; pursuing performance breakthroughs in key areas; using advanced change techniques in support of major initiatives; establishing a pattern of executive sponsorship for all initiatives; and building future capability and capacity.

The Human Performance System is designed to attract, develop, and retain the most talented people. The idea is to hire the best people and help them develop their skills, talents, and knowledge over time. Of course, it becomes more critical, as they add abilities and know-how, that we reward them properly so they feel good about their work and choose to remain with the organization as loyal employees.

A system of metrics, reviews, and course corrections keeps the business on track. Organizations need concrete measures that facilitate quality control, consistent behaviors, and predictable productivity and results. Within these parameters, control is instrumental to viability and profitability. Every activity has a set of daily rituals and measures. Leaders establish and maintain the measurement system to ensure disciplined processes.

They track progress against strategy and planning; review status on operational results through clear key metrics; update the strategy regularly; and ensure action is driven by insight based on relevant, current information that is focused on achieving the vision. This Six Systems frame helps people see how everything is integrated. Again, until the Leadership System operates effectively, all other systems are degraded. We work with leaders to ensure their Leadership System is highly effective, and we have dozens of cases that demonstrate the power of using a Whole Systems Approach.

These transformation efforts were not only successful, but more importantly, the success was sustained over time. Sadly, we also witnessed transformation efforts that were less than successful and in some cases failed. These failures could be linked directly to a failure of leadership to consciously transform individually and collectively. Anderson and William A. Adams Wiley, They are coauthors of Mastering Leadership Wiley.

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