One-sided Surfaces Research Articles

Essential surfaces in Seifert fiber spaces with singular surfaces

Two-sided incompressible surfaces in Seifert fiber spaces with isolated singular fibers are well-understood. Frohman [3] and Rannard [6] have shown that one-sided incompressible surfaces in Seifert fiber spaces which have isolated singular fibers are either pseudo-horizontal or psuedo-vertical. We extend their result to characterise essential surfaces in Seifert fiber spaces which may have singular surfaces, i.e., in S1-foliated 3-manifolds which have fibered model neighbourhoods that are isomorphic to either a fibered solid torus or a fibered solid Klein bottle.

К вопросу о классификации аналитических поверхностей

Until this day, in scientific and technical literature there are descriptions of around 600 analytical surfaces, divided into 38 classes. In turn, it is possible to divide almost each class of surfaces into sub-classes, groups, or types, that is, to compose a classification for surfaces. It will make it easier their further study. For some classes of surfaces there is no sense to set up classification schemes, since these classes consist of a small number of surfaces, or the classification will contain a bare listing of surfaces. These surfaces can include minimal and one-sided surfaces, spiral and helical ones, surfaces of Ciceika and Edlinger, as well as undulating, wavy and corrugated surfaces, and several others. The "others" category includes surfaces which, although highlighted in separate classes, are few, or they are in the process of studying. The study of scientific literature has shown that a classification for revolution surfaces, as well as for transfer and Peterson ones has not yet been set up. This paper is devoted to these classes of analytical surfaces. As the study has showed, it is possible to draw interesting conclusions based on presented classifications. In this paper it has been shown how the appearance of scientific publications devoted to classification for surfaces promote the emergence of numerous works on application of relevant surfaces in architecture, construction, engineering, shipbuilding, light industry products, topography and cartography. But devotion to analytic surfaces’ geometric forms should not be an end in itself in actual practice. So, today architects are passionate about parametric architecture, which is based on knowledge of analytical surfaces’ geometry. However, some leading architects believe that the parametric architecture oversaturated the human life’s space, resulting in a style crisis. Despite architects and engineers’ conflicting opinions, geometers must go ahead of society's immediate demands, creating new forms, classifications, and proposals.

Geometry and quantum physics in the design of an electronic device

The article considers a scientific problem in quantum mechanics related to geometry. The famous German mathematician Mobius A.F. has opened a new page in geometry — one-sided surfaces. The first of these surfaces was named after him: a ribbon (or sheet) Mobius. Almost simultaneously with Mobius, another German mathematician, Listing I., discovered this surface. Later, other one-sided surfaces were discovered. However, it turned out that they are a modification of the Mobius sheet. An example is given of how a modern engineer solved a technical problem in an electronic device by accidentally recalling a Mobius tape. Physicists are still trying to scientifically explain the positive effect in the operation of this electronic device. The observed physical phenomenon is called the Einstein-Podolsky-Rosen paradox

Prospects for the use of one-sided surfaces in design practice

The article provides examples of using one-sided surfaces, which are completed static structures with surfaces turned inside out, in design practice. One-sided surfaces are the objects of study of topological mathematicians, but in recent years these surfaces have been employed in their research by physicists and chemists. One-sided surfaces include the following surfaces: the Mobius strip, the Klein bottle, the heptahedron and the Steiner surface. One-sided surfaces were discovered and studied by German mathematicians A. Mobius and J. Listing independently of each other in the middle of the 19ᵗʰ century. The first patents for the engineers» inventions, which used the properties of one-sided surfaces, appeared in the 20ᵗʰ century. The latest scientists» achievements in the field of physics and chemistry, awarded the Nobel Prize, were obtained by using the properties of one-sided surfaces. The material of the article will allow drawing the attention of design engineers to one-sided surfaces when creating new machines and improving technological processes. Layouts of these surfaces are printed on a 3D printer using electronic models of the Mobius strip, the Klein bottle and the heptahedron.

Optimization of thermal regimes of microwave resonators digital active phased array antenna

Objective. The aim of the study is to simulate resonators for microwave digital active phased antenna arrays (DFAAR) in the microwave range, which will optimize the parameters of the antenna system for more reliable direction finding with high reliability of any aircraft. Method. The use of one-sided surfaces in the Mobius shape as resonators makes it possible to implement a new type of microwave resonators in the form of an integrated capacitance and a two-turn inductance. The parameters of such a resonator make it possible to realize a traveling or standing wave, depending on the geometric dimensions. Result. The use of innovative resonators for microwave DFAAR with p-i-n diodes to change the area of the capacitor makes it possible to implement a two-frequency method of pulse-code modulation. On the basis of such resonators, it is possible to realize a standing wave at one position of the p-i-n diodes, and a traveling wave at the other. Such electromagnetic waves, transmitted by the DFAAR, can track objects in the stealth mode. Conclusion. The use of traditional resonators for DFAAR of the microwave range allows you to discretely adjust the parameters of these resonators by turning on / off p-i-n diodes with positive and negative voltages to control the frequency properties of the resonators and set the traveling or standing wave modes.

Application of 3d technologies for microwave range electronic components creating

This paper presents the results of the study of microwave radio-electronic components electrophysical parameters manufactured on a 3D printer. Laser sintering technologies for metal semiconductor and ceramic powders were used as experimental research methods. The resonant properties of one-sided surfaces in the form of a Mobius sheet made on the 3D printer are analyzed. An explanation of a traveling and standing wave formation in the oscillatory contours of the microwave range of one-sided surfaces in the form of a Mobius sheet is presented. The possibility to create digital active phased antenna arrays based on onesided surfaces in the form of a Mobius sheet is shown.

Эволюция объектов параметризма

Modern trends in parametric architecture take us away from the use of static surfaces, pushing the dynamics to the first place. A new approach to architecture is opened, based on the dynamic development of shaping. The desire to develop the most optimal ratio of strength and expended material leads to an assessment of the prospects for the use of minimal and one-sided surfaces. Using the example of varying the parameters of a onesided Klein surface, the evolution of a parametric object from a simple Mobius strip to a complex dynamic form is demonstrated. The formation of the analytical surface in various software packages was studied. The sphere of applicability of various stages of form evolution to architectural objects is established.

On a Torus Model

The surface of parallel transfer is a surface that admits the parametrization r(u,v) = U(u) + V(v). The surface of parallel transfer in E3 can be considered as a surface produced by parallel transfer of one line along the other.In this study the torus M is different from the classic torus T, which is obtained by rotating the circle along the axis. We consider the torus as the surface of parallel transfer obtained by parallel transfer of one circle along the other. The circles are located in mutually orthogonal intersecting planes.The closed curve on the torus M is defined with the 4n-periodic vector-function.Using this function, the equations of one-sided surfaces are obtained. In particular, the equation of the Mobius band with the mentioned curve being the boundary is found.The Klein bottle is cut into two Moebius band. Also, the cross-cap is considered.The inversion of torus T and the inversion of torus M is studied in this paper.The studied surfaces are constructed in Euclidean space E3 with the help of a mathematical software toolkit.DOI 10.14258/izvasu(2018)1-24

Subjective reality and brain topology: Inversion transformations on non-orientable atomic surfaces of membrane channels.

Subject-object relations reflect the relation of phenomenology and physics and are at the centre of interest in brain research and neuro-psychology. The unresolved dichotomy behind this relation is one of the most challenging questions of our time. Setting out from causal modelling I suggest a particular topology for subject-object relations and argue that we can find a physical realization in living organism that provides a continuous transform between both domains. In a geometrical metaphor this transform has the topological properties of a one-sided surface or non-orientable flat. I argue that such a surface can be found within the electronic organization of atomic linings in the filter region of ion-conducting membrane proteins. Electron transfer along these atomic surfaces makes chiral induced spin changes to a promising signature of subject-object relations and has found experimental evidence in previous studies. I finally advocate the view that there is a basic dualism between subject and object which is physical on both sides and realized by an inversion relation along one-sided surfaces. The transition between these two aspects however is non-physical and hosts the phenomenology that characterizes subjectivity.

Тор Клиффорда и бутылка Клейна

A surface is called the one-sided surface if a local orientation along a closed curve on a tangent space changes its sign. The simplest one-sided surface is the Mobius strip. The Klein bottle and cross-cap are also one-sided surfaces. The Klein bottle can be considered as two Mobius strips glued on edges. In this paper, the Klein bottle is cut into two Mobius strips. We define the closed curve on the Clifford torus in E4 using 4π-periodic vector-function. We get a cross-cap if the Mobius strip degenerates into a point. Indicatrices of normal curvature for the investigated surfaces are studied with the help of computer mathematics. For the Klein bottle, the indicatrix of normal curvature is an ellipse. If the Klein bottle degenerates into a cross-сap, then the indicatrix of normal curvature is a circle.

Second variation of one-sided complete minimal surfaces

The stability and the index of complete one-sided minimal surfaces of certain 3-dimensional Riemannian manifolds with positive scalar curvature are studied.

INCOMPRESSIBLE ONE-SIDED SURFACES IN EVEN FILLINGS OF FIGURE 8 KNOT SPACE

In the closed, non-Haken, hyperbolic class of examples generated by (2p, q) Dehn fillings of Figure 8 knot space, the geometrically incompressible one-sided surfaces are identified by the filling ratio [Formula: see text] and determined to be unique in all cases. When applied to one-sided Heegaard splitting, this can be used to classify all geometrically incompressible splittings in this class of closed, hyperbolic examples; no analogous classification exists for two-sided Heegaard splitting.

Möbius bands of wood and alabaster

Möbius bands, those mysterious one-sided surfaces, can be beautiful objects of art when carved as three-dimensional forms in exotic wood or alabaster. The realizations of Möbius bands by a sculptor sharply contrast with their various realizations by mathematicians: as abstract geometric forms or surfaces in 3-space described by equations.

Topology in Chemistry: Designing Möbius Molecules

In 1747, Johann Sebastian Bach wrote a series of canons and fugues on the occasion of his visit to King Frederick the Great of Prussia in Potsdam. His music of homage, the “Musikalisches Opfer” (Musical Offering, BWV1079), includes a canon that is probably one of the oldest man-made examples of an object with Mobius topology.1 The canon cancricans (crab canon) is written for two violins playing the same score in two different directions. If you cut the score, paste it in such way that it forms a band with the notes of the first violin on top and those of the second violin below, fold it lengthwise between the two notes, give one end a half twist, and join both ends, you end up with a endless Mobius-shaped score (Figure 1). After every turn the two violins change their parts. Karl Amadeus Hartmann, Nicolas Slonimsky,2 and others several centuries later wrote further examples of Mobius music. Note that there are two different ways to form the Mobius score, by twisting the strip either clockwise or anticlockwise. The two Mobius strips are enantiomers (the music, unfortunately, is not) and exhibit the highest symmetry a Mobius object can attain: C2. More than 100 years later, August Ferdinand Mobius and Johann Benedict Listing (Figure 2), both students of Carl Friedrich Gauss in Gottingen, published their seminal work on projective planes and one-sided surfaces. Nowadays, the credit goes almost exclusively to Mobius3 after whom the famous band is named, even though Listing published 4 years earlier.4 Both mathematicians mentioned the “Mobius” band in unpublished papers for the first time in 1858; however, again Listing’s unpublished note (July) predates the one of Mobius (September) by 2 months. It would probably be fair to name the one-sided band a “Listing band”. There are good reasons to assume that Listing was not adequately credited because of social rather than scientific reasons.5 Most of the objects we are dealing with in our everyday lives are two-sided. They have an inside and an outside surface like a sphere, a cube, or a torus. This is also true for most two-dimensional mathematical objects. According to the mathematical definition, a normal vector on the surface of a two-sided object cannot be shifted to any other point on the surface without crossing a border or interpreted in anthropomorphic terms: walking upright on the outside surface you cannot enter the inside and vice versa. The Mobius band is probably the simplest and best-known exception (Figure 3). Any closed band with an odd number of 180° twists is one-sided, and those with an even number of 180° twists are two-sided. Unfortunately, the sidedness is not an intrinsic property, because it requires that the surface is embedded in a † In memory of Professor Edgar Heilbronner. * To whom correspondence should be addressed. Telephone: ++49 431 88

One-sided complete stable minimal surfaces

We prove that there are no complete one-sided stable minimal surfaces in the Euclidean 3-space. We classify least area surfaces in the quotient of $\mathbb{R}^3$ by one or two linearly independent translations and we give sharp upper bounds of the genus of compact two-sided index one minimal surfaces in non-negatively curved ambient spaces. Finally we estimate from below the index of complete minimal surfaces in flat spaces in terms of the topology of the surface

Some properties of an open‐ended circular waveguide with one‐ and two‐sided ideal hard walls

AbstractThe problem of radiation from a hard open‐ended circular waveguide excited in the dominant TEM mode is solved. The cases of one‐sided and two‐sided hard‐wall surfaces are considered. The rigorous solution obtained shows that the reflected TEM mode field in the latter case, unlike that in the former case, is completely concentrated at the surface outside the waveguide. © 2004 Wiley Periodicals, Inc. Microwave Opt Technol Lett 43: 160–164, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.20407

Fragments of Infinity: A Kaleidoscope of Math and Art

A visual journey to the intersection of math and imagination, guided by an award-winning author Mathematics is right brain work, art left brain, right? Not so. This intriguing book shows how intertwined the disciplines are. Portraying the work of many contemporary artists in media from metals to glass to snow, Fragments of Infinity draws us into the mysteries of one-sided surfaces, four-dimensional spaces, self-similar structures, and other bizarre or seemingly impossible features of modern mathematics as they are given visible expression. Featuring more than 250 beautiful illustrations and photographs of artworks ranging from sculptures both massive and minute to elaborate geometric tapestries and mosaics of startling complexity, this is an enthralling exploration of abstract shapes, space, and time made tangible. Ivars Peterson (Washington, DC) is the mathematics writer and online editor of Science News and the author of The Jungles of Randomness (Wiley: 0-471-16449-6), as well as four previous trade books.

One-sided incompressible surfaces in Seifert fibered spaces

The author proves a structure theorem for one-sided incompressible surfaces in Seifert fibered spaces, and studies the isotopy problem for a class of surfaces (called pseudovertical surfaces) that are supplied by the structure theorem. He then uses the structure theorem to solve the existence and homeomorphism problem for one-sided incompressible surfaces in circle bundles over a closed orientable surface.

Nonorientable surfaces in some non-Haken 3-manifolds

If a closed, irreducible, orientable 3 3 -manifold M M does not possess any 2 2 -sided incompressible surfaces, then it can be very useful to investigate embedded one-sided surfaces in M M of minimal genus. In this paper such 3 3 -manifolds M M are studied which admit embeddings of the nonorientable surface K K of genus 3 3 . We prove that a 3 3 -manifold M M of the above type has at most 3 3 different isotopy classes of embeddings of K K representing a fixed element of H 2 ( M , Z 2 ) {H_2}(M,\,{Z_2}) . If M M is either a binary octahedral space, an appropriate lens space or Seifert manifold, or if M M has a particular type of fibered knot, then it is shown that the embedding of K K in M M realizing a specific homology class is unique up to isotopy.

One-Sided Surfaces and Orientability

John W. Woll, Jr., is a Professor and Graduate Advisor at Western Washington State College in Bellingham, Washington. He has published several articles in physics and mathematics journals and is the author of a textbook on functions of several variables, published in 1964. In this article we discuss the concepts of orientability and sidedness and show that they are not the same. In fact we will show that a 2-torus imbedded in the right 3-manifold may have only one side, whereas objects like the Klein bottle and the Mobius band, which are generally considered to be one sided, may have two sides. We will be able to demonstrate, by using the classification scheme for 2-manifolds, that the 2-sphere is the only 2-manifold which does not admit both one-sided and two-sided imbeddings in a 3-manifold. The corresponding statement is not true in higher dimensions, and we reduce the problem of the number of sides an n-manifold may have when imbedded in an (n+ 1)-manifold to a question in group theory. The article was designed to (1) display on an elementary level the possibilities of cross-fertilization between different mathematical disciplines, in this case topology and group theory; (2) serve as an eye opener to the rich structure of manifolds; and (3) introduce selected topics from algebraic topology. As a consequence all concepts and arguments used have a high intuitive appeal, and there is almost an overabundance of examples, especially of 3-manifolds. It is pitched at an introductory survey level and, except for a few points, no outside information is used. The pace is leisurely until the end where it is stepped up a trifle.