Введение
Половинная модель перевернутой сферы
The Morin surface is the half way model of the sphere eversion discovered by Bernard Morin. It features fourfold rotational symmetry. If the original sphere to be everted has its outer surface colored green and its inner surface colored red, then when the sphere is transformed through homotopy into a Morin surface, half of the outwardly visible Morin surface will be green, and half red:
Half of a Morin surface corresponds to the exterior (green) of the sphere to which it is homeomorphic, and the other symmetric half to the interior (red). Then, rotating the surface 90° around its axis of symmetry will exchange its colors, i. e. will exchange the inner outer polarity of the orientable surface, so that retracing the steps of the homotopy at exactly the same position back to the original sphere after having so rotated the Morin surface will yield a sphere whose outer surface is red and whose inner surface is green: a sphere which has been turned inside out. The following is a summary of the eversion:
1. sphere: green outside, red inside
2. transforms into
3. Morin surface,
3'. Morin surface rotated 90°
2'. inversely transforms into
1'. sphere: red outside, green inside.
Поверхность Морина — это промежуточная модель эверсии сферы, открытая Бернаром Морином. Она обладает четырехкратной вращательной симметрией. Если исходная сфера, подлежащая эверсии, имеет зеленую внешнюю поверхность и красную внутреннюю поверхность, то при преобразовании сферы посредством гомотопии в поверхность Морина, половина внешней видимой поверхности Морина будет зеленой, а другая половина — красной: половина поверхности Морина соответствует внешней (зеленой) стороне сферы, к которой она гомеоморфна, а другая симметричная половина — внутренней (красной). Затем, вращение поверхности на 90° вокруг её оси симметрии меняет цвета местами, то есть меняет внутреннюю и внешнюю полярность ориентируемой поверхности. Таким образом, повторение шагов гомотопии в точно таком же положении обратно к исходной сфере после такого вращения поверхности Морина приведет к сфере, у которой внешняя поверхность красная, а внутренняя — зеленая: сфера, вывернутая наизнанку. Ниже приведено краткое описание эверсии:
The Morin surface is the half way model of the sphere eversion discovered by Bernard Morin. It features fourfold rotational symmetry. If the original sphere to be everted has its outer surface colored green and its inner surface colored red, then when the sphere is transformed through homotopy into a Morin surface, half of the outwardly visible Morin surface will be green, and half red:
Half of a Morin surface corresponds to the exterior (green) of the sphere to which it is homeomorphic, and the other symmetric half to the interior (red). Then, rotating the surface 90° around its axis of symmetry will exchange its colors, i. e. will exchange the inner outer polarity of the orientable surface, so that retracing the steps of the homotopy at exactly the same position back to the original sphere after having so rotated the Morin surface will yield a sphere whose outer surface is red and whose inner surface is green: a sphere which has been turned inside out. The following is a summary of the eversion:
1. sphere: green outside, red inside
2. transforms into
3. Morin surface,
3'. Morin surface rotated 90°
2'. inversely transforms into
1'. sphere: red outside, green inside.
1. сфера: зеленая снаружи, красная внутри
2. преобразуется в
3. поверхность Морина,
3'. поверхность Морина, повернутая на 90°
2'. обратное преобразование в
1'. сфера: красная снаружи, зеленая внутри.
The Morin surface is the half way model of the sphere eversion discovered by Bernard Morin. It features fourfold rotational symmetry. If the original sphere to be everted has its outer surface colored green and its inner surface colored red, then when the sphere is transformed through homotopy into a Morin surface, half of the outwardly visible Morin surface will be green, and half red:
Half of a Morin surface corresponds to the exterior (green) of the sphere to which it is homeomorphic, and the other symmetric half to the interior (red). Then, rotating the surface 90° around its axis of symmetry will exchange its colors, i. e. will exchange the inner outer polarity of the orientable surface, so that retracing the steps of the homotopy at exactly the same position back to the original sphere after having so rotated the Morin surface will yield a sphere whose outer surface is red and whose inner surface is green: a sphere which has been turned inside out. The following is a summary of the eversion:
1. sphere: green outside, red inside
2. transforms into
3. Morin surface,
3'. Morin surface rotated 90°
2'. inversely transforms into
1'. sphere: red outside, green inside.
Галерея поверхности Морина
Четыре различных вида поверхности Морина: первые два показаны с вырезанными "проходными барьерами", последние два – вид снизу.
Аналитическая поверхность Морина
Поверхность Морина может быть элегантно описана набором уравнений как в открытой форме (с полюсами, уходящими в бесконечность), так и в замкнутой.