An analogue of a theorem of Steinitz for ball polyhedra in $$\mathbb {R}^3$$

On a 3-Dimensional Isoperimetric Problem

Canadian Mathematical Bulletin ◽

10.4153/cmb-1970-083-4 ◽

1970 ◽

Vol 13 (4) ◽

pp. 447-449 ◽

Cited By ~ 3

Author(s):

Magelone Kömhoff

Keyword(s):

Surface Area ◽

Convex Polyhedron ◽

Three Dimensional ◽

Isoperimetric Problem ◽

Total Surface Area ◽

Edge Length ◽

Regular Tetrahedron ◽

3 Dimensional ◽

Image Position

Let L(P) denote the total edge length and A(P) the total surface area of a three-dimensional convex polyhedron P. In [5] it was shown that if P belongs to the set of all polyhedra with triangular faces then for all with equality if and only if is a regular tetrahedron.It is not difficult to establish the inequality

Ordering of convex polyhedra and the Fedorov algorithm

Acta Crystallographica Section A Foundations and Advances ◽

10.1107/s2053273316017095 ◽

2017 ◽

Vol 73 (1) ◽

pp. 77-80 ◽

Cited By ~ 2

Author(s):

Yury L. Voytekhovsky

Keyword(s):

Adjacency Matrix ◽

Convex Polyhedron ◽

Convex Polyhedra ◽

Numerical Code ◽

Edge Graph ◽

The Relationship

A method of naming any convex polyhedron by a numerical code arising from the adjacency matrix of its edge graph has been previously suggested. A polyhedron can be built using its name. Classes of convexn-acra (i.e.n-vertex polyhedra) are strictly (without overlapping) ordered by their names. In this paper the relationship between the Fedorov algorithm to generate the whole combinatorial variety of convex polyhedra and the above ordering is described. The convexn-acra are weakly ordered by the maximum extra valencies of their vertices. Thus, non-simplen-acra follow the simple ones for anyn.

How to name and order convex polyhedra

Acta Crystallographica Section A Foundations and Advances ◽

10.1107/s2053273316010846 ◽

2016 ◽

Vol 72 (5) ◽

pp. 582-585 ◽

Cited By ~ 5

Author(s):

Yury L. Voytekhovsky

Keyword(s):

Adjacency Matrix ◽

Convex Polyhedron ◽

Convex Polyhedra ◽

Numerical Code ◽

Edge Graph

In this paper a method is suggested for naming any convex polyhedron by a numerical code arising from the adjacency matrix of its edge graph. A polyhedron is uniquely fixed by its name and can be built using it. Classes of convexn-acra (i.e.n-vertex polyhedra) are strictly ordered by their names.

On the complements of 3-dimensional convex polyhedra as polynomial images of ℝ3

International Journal of Mathematics ◽

10.1142/s0129167x14500712 ◽

2014 ◽

Vol 25 (07) ◽

pp. 1450071 ◽

Cited By ~ 5

Author(s):

José F. Fernando ◽

Carlos Ueno

Keyword(s):

Convex Polyhedron ◽

Convex Polyhedra ◽

Semialgebraic Sets ◽

3 Dimensional

Let [Formula: see text] be a convex polyhedron of dimension n. Denote [Formula: see text] and let [Formula: see text] be its closure. We prove that for n = 3 the semialgebraic sets [Formula: see text] and [Formula: see text] are polynomial images of ℝ3. The former techniques cannot be extended in general to represent the semialgebraic sets [Formula: see text] and [Formula: see text] as polynomial images of ℝn if n ≥ 4.

Three-Dimensional Reconstruction Using Stereo Pairs from Serial Thick Sections

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100080249 ◽

1977 ◽

Vol 35 ◽

pp. 562-563

Author(s):

Robert Glaeser ◽

Thomas Bauer ◽

David Grano

Keyword(s):

Three Dimensional ◽

Dimensional Structure ◽

Serial Sections ◽

Thin Sections ◽

3 Dimensional ◽

Transmission Electron ◽

Freeze Dried ◽

Dimensional Reconstruction ◽

Stereo Imagery ◽

Whole Mounts

In transmission electron microscopy, the 3-dimensional structure of an object is usually obtained in one of two ways. For objects which can be included in one specimen, as for example with elements included in freeze- dried whole mounts and examined with a high voltage microscope, stereo pairs can be obtained which exhibit the 3-D structure of the element. For objects which can not be included in one specimen, the 3-D shape is obtained by reconstruction from serial sections. However, without stereo imagery, only detail which remains constant within the thickness of the section can be used in the reconstruction; consequently, the choice is between a low resolution reconstruction using a few thick sections and a better resolution reconstruction using many thin sections, generally a tedious chore. This paper describes an approach to 3-D reconstruction which uses stereo images of serial thick sections to reconstruct an object including detail which changes within the depth of an individual thick section.

Structural preservation and absolute contrast of catalase crystal sections prepared with tannic acid

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s042482010007597x ◽

1983 ◽

Vol 41 ◽

pp. 448-449

Author(s):

C.W. Akey ◽

M. Szalay ◽

S.J. Edelstein

Keyword(s):

Tannic Acid ◽

Beef Heart ◽

X Rays ◽

Screw Axis ◽

General Applicability ◽

Thin Sections ◽

Beef Liver ◽

3 Dimensional ◽

Dimensional Reconstruction ◽

Structural Preservation

Three methods of obtaining 20 Å resolution in sectioned protein crystals have recently been described. They include tannic acid fixation, low temperature embedding and grid sectioning. To be useful for 3-dimensional reconstruction thin sections must possess suitable resolution, structural fidelity and a known contrast. Tannic acid fixation appears to satisfy the above criteria based on studies of crystals of Pseudomonas cytochrome oxidase, orthorhombic beef liver catalase and beef heart F1-ATPase. In order to develop methods with general applicability, we have concentrated our efforts on a trigonal modification of catalase which routinely demonstrated a resolution of 40 Å. The catalase system is particularly useful since a comparison with the structure recently solved with x-rays will permit evaluation of the accuracy of 3-D reconstructions of sectioned crystals.Initially, we re-evaluated the packing of trigonal catalase crystals studied by Longley. Images of the (001) plane are of particular interest since they give a projection down the 31-screw axis in space group P3121. Images obtained by the method of Longley or by tannic acid fixation are negatively contrasted since control experiments with orthorhombic catalase plates yield negatively stained specimens with conditions used for the larger trigonal crystals.

Deviations from Ideal Decagonal Symmetry in Electron Diffraction Patterns of the “Decagonal” Phase in the Al-Co-Cu Alloy System

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100088889 ◽

1991 ◽

Vol 49 ◽

pp. 914-915

Author(s):

Atul S. Ramani ◽

Earle R. Ryba ◽

Paul R. Howell

Keyword(s):

Electron Microscope ◽

Alloy System ◽

Nominal Composition ◽

Dimensional Lattice ◽

3 Dimensional ◽

Decagonal Phase ◽

Cu Alloy ◽

Transmission Electron ◽

Lattice Periodicity ◽

Diffraction Patterns

The “decagonal” phase in the Al-Co-Cu system of nominal composition Al65CO15Cu20 first discovered by He et al. is especially suitable as a topic of investigation since it has been claimed that it is thermodynamically stable and is reported to be periodic in the dimension perpendicular to the plane of quasiperiodic 10-fold symmetry. It can thus be expected that it is an important link between fully periodic and fully quasiperiodic phases. In the present paper, we report important findings of our transmission electron microscope (TEM) study that concern deviations from ideal decagonal symmetry of selected area diffraction patterns (SADPs) obtained from several “decagonal” phase crystals and also observation of a lattice of main reflections on the 10-fold and 2-fold SADPs that implies complete 3-dimensional lattice periodicity and the fundamentally incommensurate nature of the “decagonal” phase. We also present diffraction evidence for a new transition phase that can be classified as being one-dimensionally quasiperiodic if the lattice of main reflections is ignored.

A Novel Reconstitution Method for Inducing the Formation of Regular 2-Dimensional Arrays of Membrane Proteins and Lipids

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100102845 ◽

1988 ◽

Vol 46 ◽

pp. 152-153 ◽

Cited By ~ 1

Author(s):

A. Engel ◽

A. Holzenburg ◽

K. Stauffer ◽

J. Rosenbusch ◽

U. Aebi

Keyword(s):

Membrane Proteins ◽

Flow Rate ◽

Lipid Membranes ◽

Functional Characterization ◽

Dimensional Structure ◽

Electron Crystallography ◽

Peltier Element ◽

Protein Ratio ◽

Precise Control ◽

3 Dimensional

Reconstitution of solubilized and purified membrane proteins in the presence of phospholipids into vesicles allows their functions to be studied by simple bulk measurements (e.g. diffusion of differently sized solutes) or by conductance measurements after transformation into planar membranes. On the other hand, reconstitution into regular protein-lipid arrays, usually forming at a specific lipid-to-protein ratio, provides the basis for determining the 3-dimensional structure of membrane proteins employing the tools of electron crystallography.To refine reconstitution conditions for reproducibly inducing formation of large and highly ordered protein-lipid membranes that are suitable for both electron crystallography and patch clamping experiments aimed at their functional characterization, we built a flow-dialysis device that allows precise control of temperature and flow-rate (Fig. 1). The flow rate is generated by a peristaltic pump and can be adjusted from 1 to 500 ml/h. The dialysis buffer is brought to a preselected temperature during its travel through a meandering path before it enters the dialysis reservoir. A Z-80 based computer controls a Peltier element allowing the temperature profile to be programmed as function of time.

Structure and Interaction of Protamine by Dark Field Electron Microscopy

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100090786 ◽

1976 ◽

Vol 34 ◽

pp. 160-161

Author(s):

D.P. Bazett-Jones ◽

F.P. Ottensmeyer

Keyword(s):

Electron Microscopy ◽

Dark Field ◽

Amino Acid Sequences ◽

Dimensional Structure ◽

3 Dimensional ◽

Field Electron ◽

Proposed Model ◽

Field Electron Microscopy ◽

Dark Field Electron ◽

Positively Charged

It has been shown for some time that it is possible to obtain images of small unstained proteins, with a resolution of approximately 5Å using dark field electron microscopy (1,2). Applying this technique, we have observed a uniformity in size and shape of the 2-dimensional images of pure specimens of fish protamines (salmon, herring (clupeine, Y-l) and rainbow trout (Salmo irideus)). On the basis of these images, a model for the 3-dimensional structure of the fish protamines has been proposed (2).The known amino acid sequences of fish protamines show stretches of positively charged arginines, separated by regions of neutral amino acids (3). The proposed model for protamine structure (2) consists of an irregular, right-handed helix with the segments of adjacent arginines forming the loops of the coil.

Design and calibration of an IVEM for general 3-dimensional imaging of non-diffracting materials

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100129838 ◽

1992 ◽

Vol 50 (2) ◽

pp. 1040-1041

Author(s):

Neil Rowlands ◽

Jeff Price ◽

Michael Kersker ◽

Seichi Suzuki ◽

Steve Young ◽

...

Keyword(s):

3D Imaging ◽

Tomographic Reconstruction ◽

Three Dimensional ◽

3 Dimensional ◽

3D Microstructure ◽

Image Translation ◽

Digital Correlation ◽

On Line ◽

Mechanical Stage ◽

Projection Angle

Three-dimensional (3D) microstructure visualization on the electron microscope requires that the sample be tilted to different positions to collect a series of projections. This tilting should be performed rapidly for on-line stereo viewing and precisely for off-line tomographic reconstruction. Usually a projection series is collected using mechanical stage tilt alone. The stereo pairs must be viewed off-line and the 60 to 120 tomographic projections must be aligned with fiduciary markers or digital correlation methods. The delay in viewing stereo pairs and the alignment problems in tomographic reconstruction could be eliminated or improved by tilting the beam if such tilt could be accomplished without image translation.A microscope capable of beam tilt with simultaneous image shift to eliminate tilt-induced translation has been investigated for 3D imaging of thick (1 μm) biologic specimens. By tilting the beam above and through the specimen and bringing it back below the specimen, a brightfield image with a projection angle corresponding to the beam tilt angle can be recorded (Fig. 1a).