Working with Siebert, I have developed a program (colloquially referred to as the Gross-Siebert program) for exploring the underlying geometry of mirror symmetry using methods from algebraic and tropical geometry. Mathematical explorations of mirror symmetry have led to deep and profound insights in algebraic, logarithmic, symplectic and differential geometry, as well as algebraic combinatorics and representation theory. This relationship can be summarized by the rough statement that the symplectic geometry of X is isomorphic to the complex geometry of Y. Further, mirror symmetry posits an intricate relationship between the geometry of the members of the pair. This phenomenon, when described in mathematical language, posits that certain kinds of geometric objects, known as Calabi-Yau manifolds, come in pairs X, Y. The “metry” in symmetry and geometry and metric refers to measurement.Mirror symmetry is a phenomenon first discovered by string theorists in 1989.
The Mirror Puzzle Book, by Marion Walter What’s in a word? Make a Bigger Puddle Make a Smaller Worm, by Marion Walter Any rotation of any amount around the center of the circle also leaves the circle unchanged. The most symmetric shapeĪ circle has infinitely many lines of symmetry: any diameter lies on a line of symmetry through the center of the circle. Note that some figures, like the star and the colorful blob at the top of the page, but not the letters N, Z, or S, have both reflective and rotational symmetry. The letters, N, Z, and S also share that property.
For example, these figures, when rotated just the right amount - 360°/3 for the “name” picture and 360°/5 for the star - look precisely as they did before rotation.
#LOGO WITH A REFLECTION SYMMETRY FULL#
Rotational symmetryĪnother symmetry that children sometimes use in their Pattern Block designs is “Rotational Symmetry.” A figure has rotational symmetry if some rotation (other than a full 360° turn) produces the same figure. Some letters, for example, X, H, and O, have both vertical and horizontal lines of symmetry.Īnd some, like P, R, and N, have no lines of symmetry. Letters like B and D have a horizontal line of symmetry: their top and bottom parts match. More intrepid experiments give other interesting results. The first figure below shows that the letter A has a vertical line of symmetry, but it’s rather “tame” play. The colorful design above has only vertical and horizontal lines of symmetry, but placing a mirror on it at another angle can create a beautiful new design. Well before children begin any formal study of symmetry, playing with mirrors - perhaps on Pattern Block designs that they build - develops experience and intuition that can serve both their geometric thinking and their artistic ideas. This new shape - the combination of the triangular half of the original rectangle and its image in the mirror - is called a kite. When a mirror is placed along the diagonal of a rectangle, the result does not look the same as the original rectangle, so the diagonal is not a line of symmetry. For example, the diagonal of a (non-square) rectangle is not a line of symmetry. The star below has 5 lines of symmetry, five lines on which it can be folded so that both sides match perfectly.Ī common misconception found even in many glossaries and texts: Not all lines that divide a figure into two congruent parts are lines of symmetry. This figure has two lines of symmetry: the horizontal line of symmetry cuts the figure into a top and bottom that are mirror images of each other the vertical line of symmetry cuts the figure into a left and right that are mirror images of each other. Mirror-symmetric objects have at least one line of symmetry, a line along which the figure can be folded into two precisely matching parts, parts that are mirror images of each other. Reflective symmetry and “line of symmetry” We say that the original figure is “symmetric” with respect to the mirror it has reflective symmetry. The result that one sees - half of the original and the mirror image of that half - exactly matches the original figure. This photograph shows a simple picture with a mirror placed along the line of symmetry. This article will focus on that one meaning, but illustrate others as well. As a result, school materials tend to use the word symmetry as if it had only that one meaning. Although there are many kinds of symmetry, elementary school generally presents only reflective symmetry (or “mirror symmetry”).
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