One-sided Surface Research Articles

The Shape of a Möbius Strip

The Mobius strip, obtained by taking a rectangular strip of plastic or paper, twisting one end through 180 degrees, and then joining the ends, is the canonical example of a one-sided surface. Finding its characteristic developable shape has been an open problem ever since its first formulation in refs 1,2. Here we use the invariant variational bicomplex formalism to derive the first equilibrium equations for a wide developable strip undergoing large deformations, thereby giving the first non-trivial demonstration of the potential of this approach. We then formulate the boundary-value problem for the Mobius strip and solve it numerically. Solutions for increasing width show the formation of creases bounding nearly flat triangular regions, a feature also familiar from fabric draping and paper crumpling. This could give new insight into energy localization phenomena in unstretchable sheets, which might help to predict points of onset of tearing. It could also aid our understanding of the relationship between geometry and physical properties of nano- and microscopic Mobius strip structures.

The shape of a Möbius strip

The Möbius strip, obtained by taking a rectangular strip of plastic or paper, twisting one end through 180 degrees, and then joining the ends, is the canonical example of a one-sided surface. Finding its characteristic developable shape has been an open problem ever since its first formulation in refs 1,2. Here we use the invariant variational bicomplex formalism to derive the first equilibrium equations for a wide developable strip undergoing large deformations, thereby giving the first non-trivial demonstration of the potential of this approach. We then formulate the boundary-value problem for the Möbius strip and solve it numerically. Solutions for increasing width show the formation of creases bounding nearly flat triangular regions, a feature also familiar from fabric draping and paper crumpling. This could give new insight into energy localization phenomena in unstretchable sheets, which might help to predict points of onset of tearing. It could also aid our understanding of the relationship between geometry and physical properties of nano- and microscopic Möbius strip structures.

Constrained curve fitting on manifolds

When designing curves on surfaces the need arises to approximate a given noisy target shape by a smooth fitting shape. We discuss the problem of fitting a B -spline curve to a point cloud by squared distance minimization in the case that both the point cloud and the fitting curve are constrained to lie on a smooth manifold. The on-manifold constraint is included by using the first fundamental form of the surface for squared distance computations between the point cloud and the fitting curve. For the solution we employ a constrained optimization algorithm that allows us to include further constraints such as one-sided fitting or surface regions that have to be avoided by the fitting curve. We illustrate the effectiveness of our algorithm by means of several examples showing different applications.

Nanowires with Surface Disorder: Giant Localization Lengths and Quantum-to-Classical Crossover

We investigate electronic quantum transport through nanowires with one-sided surface roughness. A magnetic field perpendicular to the scattering region is shown to lead to exponentially diverging localization lengths in the quantum-to-classical crossover regime. This effect can be quantitatively accounted for by tunneling between the regular and the chaotic components of the underlying mixed classical phase space.

Eigenvibrations in trapped-energy contoured piezoelectric resonators with a one-sided arbitrarily oriented elliptical convexity

A mathematical model of a piezoelectric plate is discussed with a one-sided convex surface of an arbitrary shape. Generic relations are given to calculate the frequency spectrum and distributions of the vibration amplitudes with normal drive levels. A particular case of the ellipsoidal convexity is studied, that is arbitrarily oriented with respect to the plate axes. A numerical example for such a curvature is also given. Based upon this, we show that the frequency spectrum is critically dependent on the angle between the main ellipsoid axis and the rotated coordinates of a piezoelectric plate and to the ratio of the main radii of an ellipsoid. We notice that such a surface allows for a proper placement of the anharmonics in the frequency spectrum, avoiding their interaction caused by environment. It may also be useful to model the manufacturing imperfection and its influence upon the vibration spectra.

Static Strength of Mechanically Nonuniform Welded Joints with a One-Sided Surface Defect Subject to Ductile Failure

A new formula is proposed for calculation of the average limiting stress of a welded joint in a thin-wall cylindrical pipe or plate; this formula takes into account the magnitude of mechanical nonuniformity, the dimensions of the softened interlayer, and the depth and width of a surface defect located in the zone of thermal influence of the welded seam. The range of relative depths of a surface defect in a welded joint in pipes, for which the welded joint has a strength equal to the base metal, is determined theoretically.

Magnetic-field effects on the transport properties of one-sided rough wires

We present a detailed numerical analysis of the effect of a magnetic field on the transport properties of a ``small-N'' one-sided surface disordered wire. When time-reversal symmetry is broken due to a magnetic field B, we find a strong increase with B not only of the localization length $\ensuremath{\xi}$ but also of the mean free path l caused by boundary states. Despite this, the universal relationship between l and $\ensuremath{\xi}$ does hold. We also analyze the conductance distribution at the metal-insulator crossover, finding very good agreement with random matrix theory with two fluctuating channels within the circular orthogonal (unitary) ensemble in the absence (presence) of B.

Apse Alignment of the Uranian Rings

An explanation of the dynamical mechanism for apse alignment of the eccentric uranian rings is necessary before observations can be used to determine properties such as ring masses, particle sizes, and elasticities. The leading model (P. Goldreich and S. Tremaine 1979, Astron J. 84, 1638–1641) relies on the ring self-gravity to accomplish this task, yet it yields equilibrium masses which are not in accord with Voyager radio measurements. We explore possible solutions such that the self-gravity and the collisional terms are both involved in the process of apse alignment. We consider limits that correspond to a hot and a cold ring, and we show that pressure terms may play a significant role in the equilibrium conditions for the narrow uranian rings. In the cold ring case, where the scale height of the ring near periapse is comparable to the ring particle size, we introduce a new pressure correction pertaining to a region of the ring where the particles are locked in their relative positions and jammed against their neighbors and the velocity dispersion is so low that the collisions are nearly elastic. In this case, we find a solution such that the ring self-gravity maintains apse alignment against both differential precession ( m=1 mode) and the fluid pressure. We apply this model to the uranian α ring and show that, compared to the previous self-gravity model, the mass estimate for this ring increases by an order of magnitude. In the case of a hot ring, where the scale height can reach a value as much as 50 times the particle size, we find velocity dispersion profiles that result in pressure forces which act in such a way as to alter the ring equilibrium conditions, again leading to a ring mass increase of an order of magnitude. We find that such a velocity dispersion profile would require a different mechanism than is currently envisioned for establishing a heating/cooling balance in a finite-sized, inelastic particle ring. Finally, we introduce an important correction to the model of E. I. Chiang and P. Goldreich (2000, Astrophys. J. 540, 1084–1090.). These authors relied on collisional forces in the last ∼100 m of an ∼10 km wide ring to increase ring equilibrium masses by up to a factor of ∼100. However, their treatment of ring edges as one-sided surface density drops leads to a strong dependence of the ring mass on the adjustable parameter λ (the length scale over which the ring's optical depth drops from order unity to zero at the edge). A treatment of the ring edges that takes into account their ridgelike structure retains the increase of ring mass of the order of ∼100 for a 10 km wide ring, while exhibiting weak dependence on λ. We conclude that a modified Chiang–Goldreich model can likely account for the masses of narrow, eccentric planetary rings; however, the role of shepherd satellites both in forming ring edges and in altering the streamline precession conditions near them needs to be explored further. It is also unclear whether a fully self-consistent ring model allows for the possibility of rings with negative eccentricity gradients.

Three dimensional distribution of needle and stem surface area in a Douglas-fir.

The distribution of needles and branches in the crown of a 14-m tall Douglas-fir was investigated for the purpose of developing a three-dimensional structure for use with radiation transfer models. We found a linear relationship between the basal area of main branches originating at the bole and the total one-sided planimetric surface area of the foliage attached to each branch. A similar linear relationship was found between the branch basal area and the mass of stem material on the branch. Total silhouette area (needles plus branch material) did not vary significantly horizontally along the branch length, but stem surface area declined linearly from near the bole to the branch tip. Needle surface silhouette area and leaf area index increased along the main branch from near the bole to the branch tip. Silhouette leaf area (STAR) did not vary along the main branch. The needle area density (NAD) (m(2) m(-3)) was calculated for each of the lower 11 whorls; the vertical distribution of NAD increased from the base of the bole to the top of the crown.

Surface area of woody organs of an evergreen broadleaf forest tree in Japan and Southeast Asia

Architecture of evergreen broadleaf trees in evergreen warm-temperate and tropical forests was analyzed with a ratio (U/Ac) of total surface area of aboveground woody organs to leaf area (one-sided surface area) of each felled tree. The ratio,U/Ac, tended to decrease with the increasing ofdbh. There was little difference in a range of the ratio at eachdbh class between a warmtemperate forest and a tropical rainforest. The ratios of larger trees correlated with their relative growth rates ofdbh among similar sized trees. Canopy trees tended to stop their growth at some value of a ratio, a threshold value being about 1.5, irrespective of forest types. The threshold value showed the critical condition that annual respiration of woody organs of a tree consumed nearly all surplus production. On the basis of the pipe model, an ideal maximum tree height was considered with the ratio, and was estimated at 110 m and 70 m in a tropical rainforest and a warm-temperate forest, respectively.

Changes in Erythron Organization during Prolonged Cadmium Exposure: An Indicator of Heavy Metal Stress?

Changes in erythron organization were examined in goldfish (Carassius auratus) exposed for 3 wk to 5, 15, and 25% of 240-h Cd LC50. In a subsequent, more extended study, specimens were first rendered acutely anemic by immersion in phenylhydrazine HCl, allowed to recover in Cd-free water or in concentrations equivalent to ~5 or ~11% of LC50, and sampled after 2, 5, 8, and 11 wk. Total, dividing, and degrading or karyorrhectic cell numbers were recorded, as were cell length, one-sided surface area, and form factor, in fish exposed to ~11% LC50, total cell numbers and the incidence of cell division fell, while karyorrhexis increased. Dividing to total cell ratios declined with time; karyorrhectic to total and karyorrhectic to dividing cell ratios increased. Variations in cytomorphology suggested slower cell maturation at both metal concentrations. Detection of heavy metal stress in fishes by this form of hematological assessment appears feasible.

Maturation of the goldfish ( Carassius auratus) erythrocyte

1. 1. Cohorts of [ 3H]thymidine-labelled erythrocytes were examined over a 42-day period in goldfish ( Carassius auratus L.) recovering from phenylhydrazine HCl-induced anemia under normoxic conditions at 20+ 1°C and maintained with minimal disturbance on a high nutritional plane. 2. 2. As judged by changes in primary and derived hematological variables, maturation required 16–20 days. 3. 3. Similar estimates were obtained using cytomorphic variables obtained by image analysing methods. 4. 4. These suggest that juvenile red cells in this species can be identified on the basis of the following characteristics: major axis < 9.2 μM; one-sided surface area not greater than ~ 50 μm 2; axis ratio >0.774; form factor >0.938. 5. 5. Corresponding values for mature cells are: major axis > 11.2 μm; area >68.5 μm 2; axis ratio <0.716; form factor < 0.912. 6. 6. These criteria, with values for dividing and karyorrhectic cell numbers, offer a basis for more detailed and dynamic characterization of the erythron during response to environmental variation than has previously been possible.

Erythrodynamics in Goldfish, Carassius auratus L.: Temperature Effects

Goldfish, Carassius auratus L., were acclimated to constant (con15°, con25°, and con35° C) and diurnally cycling cyc25° ± 10° C) temperature regimes or exposed to abrupt heat shock (15° to shock25° C; 25° to shock35° C). Changes in peripheral erythrocyte populations were assessed in terms of red-cell numbers, hemoglobin, hematocrit, mean erythrocytic volume and hemoglobin, cell major and minor axes, one-sided surface area, axis ratio, form factor ([4π area]/[perimeter²]) and ³H-thymidine uptake, and distribution. Temperature evoked a multicomponent response involving increased rates of erythropoiesis, karyorrhexis, and division of circulating juvenile cells. In addition, there was evidence of accelerated maturation and release of juvenile cells from storage sites. As Hb phenotype options are influenced by conditions obtaining during cell formation and/or maturation, adjustment of isomorph abundances may be achieved with minimal concomitant increase in blood viscosity and cardiac work through partial elimination and replacement of preexisting erythrocytes.

Magnetohydrodynamic bending waves in a current sheet

The physical properties of MHD bending waves in an isothermal, compressible, low-beta, three-dimensional current sheet of finite thickness in which the magnetic field direction and strength varies are considered. The case of the wavenumber (k) to circular frequency ratio being greater than the Alfven velocity outside the layer (V sub A) corresponds to one-sided surface waves, and it is suggested that the heliospheric current sheet ripples are not this type of bending wave. The case of k/omega of less than V sub A describes the interaction of freely and obliquely propagating MHD waves with the layer, while the case of k/omega = V sub A describes an Alfven wave propagating parallel to but having no interaction with the layer. 24 references.

My Transformable Structures Based on the Mobius Strip

In 1966 I made a number of small sculptures of folded paper and cardboard. These objects, which I call desplegables (Spanish for 'unfolding'), were responsible for directing my attention to the branch of geometry called topology [1, 2]. I read about topological spaces with the hope of finding something to serve me in my art. In this quest I became acquainted with and much fascinated by the Mobius strip. It can be described as follows: A long rectangle (e.g. a strip of paper, 2 x 30 cm.) is held fixed at one end and turned through 180? at the other end. The two ends are then joined yielding a one-sided surface [1]. On such a surface, if one places one's finger at any point on it and then moves the finger along the strip, the finger will return to the starting point after traversing the surface on both sides of the untwisted strip. My first object, a folding sculpture, (1967) utilizing the idea of twisting and joining in a ring (40 cm. diameter) was made from a paper strip containing creases delineating a chain of equilateral triangles. This strip was given three twists (540?), instead of just one (180?), as in the M6bius strip. The resulting one-sided surface can be transformed, by folding, into five different 3dimensional forms and one 2-dimensional shape that has a hexagonal perimeter; hence the name that I gave to this type of construction is 'Hexaflexagono'. Art objects made of elements that can be rearranged by a viewer are called transformables [3] and paintings and sculptures of this kind have been devised by numerous artists in recent years [4]. I then wished to see whether other objects with flexion characteristics similar to those of a 'Hexaflexagono' could be made and whether other polygons might be formed. I found that they could and I devised a procedure in which the folding and twisting were varied. Although the formation of a Mobius strip represents one stage in their fabrication, the topological property of the Mobius strip is not preserved in the final folded objects. The 'Tetraflexagono', which I constructed in this way, can be arranged into a 2-dimensional square and into other forms including that of a cube. I have prepared a printed edition of a 'Tetraflexagono', with colored circular and linear stripes on the

Dressing Up Mathematics

Examining a Moebius strip (the onesided surface having a single edge, as shown below) leads rather easily to questions of whether there exist surfaces having one side and two edges or two sides and one edge. These particular questions are not difficult to resolve and, in fact, can be illustrated rather quickly.

The Sieve of Eratosthenes and the Mobius Strip

The M'obius strip is formed from a long rectangle of paper by rotating one end of the strip 1800 relative to the other end and joining the two ends together. The resulting strip is a one-sided surface. It may likewise be shown that the strip will be one-sided for all rotations which represent odd multiples of 1800, but will be two-sided for all rotations which represent even multiples of 1800. The ideal plane strip, prior to deformation into the Mobius figure, may be considered as a prism whose base has only two sides. From the practical standpoint, however, the strip has a finite thickness. This consideration forms the basis for a similar study of prisms of 3, 4, 5, * * *, N when deformed in a manner analogous to that of the M6bius strip. Suppose, for example, that the two bases of a regular triangular prism are joined after one base has been rotated about the long axis of the prism by 1200 with respect to the other base. All three of the prism are now continuous. The same will be true for a similar rotation of 2400. However, for rotations of 00 and 3600, the figure will present three sides. These are the only possibilities, since they will be repeated upon further twissting, depending upon whether the total rotation of one base relative to the other is 360 No, 360 NO+ 1200, or 360 No+2400, where N is an integer. In the case of the square prism, there may be 1, 2, or 4 sides depending upon the extent of the twist prior to the joining of the ends. If one base is rotated relative to the other base by 00 or by any multiple of 3600, four will of course be presented. If, however, the rotation is represented by 900 +360 No or 2700+360 N0, there will be only one side. For all rotations represented by 1800+360 No two will result. These possibilities will be repeated indefinitely with continued twisting of the prism prior to the joining of the ends. By a generalization of this process, the possibilities may be derived for any prism of number of S when one base is rotated by multiples of 3600/S, relative to the other base. The following array gives the number of surfaces obtained for prisms up to ten sides, when twisted by 0 (3600/S), 1. (360?/S), 2 (3600/S), * * * , S(3600/S). The first number in each horizontal represents the number of of the regular prism at 00 rotation. Successive numbers from left to right represent the number of surfaces obtained for the first, second, etc. multiples of 3600/S. It is apparent that every horizontal repeats itself both diagonally and vertically. It is also obvious that each row of the array may be generated from purely numerical considerations by setting down, from left tQ right, the highest factor

One-sided minimal surfaces with a given boundary

One-sided surfaces present themselves quite naturally in the theory of minimal surfaces; they were first studied systematically by Sophus Liet under the name of Minimaldoppelflachen. Lie had interpreted the formulas of Monge for a minimal surface in the now well known geometric way: that every minimal surface can be generated by translating one minimal curvet along another. An equivalent construction is to take the mid-points of all line segments whose ends lie respectively on any two fixed minimal curves gi, g2; their locus is a minimal surface, which is real whenever ,ut, A2 are conjugate complex. If li and A2 coincide in a minimal curve A, this construction becomes the taking of the locus of the midpoints of all chords of ,u. The resulting minimal surface, real whenever / is its own conjugate complex curve, is of the type designated as Minimaldoppelflache by Lie. If this surface is not periodic, i.e. does not go over into itself by a certain translation and its repetitions, then it is a one-sided surface in the sense of topology: a,material point moving continuously on the surface can pass from any position to that directly beneath without crossing over any boundary of the surface (M6bius strip); or, if we ascribe a certain arrow to the normal at any fixed point and follow the continuous variation of this sensed normal as the point moves on the surface, it is possible to describe a closed path which will reverse this arrow. In particular, the surface cannot be periodic if it is algebraic; therefore every algebraic double minimal surface is one-sided. Algebraic character can be secured for the surface by taking the minimal curve pu to be algebraic; this in turn can be done by taking as algebraic the arbitrary function f(t) in the Weierstrass formulas for a minimal curve:?