Polytope

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Polytope 

A polyhedron is a 3-dimensional polytope

A polygon is a 2-dimensional polytope. Polygons can be characterised according to various criteria. Some examples are: open (excluding its boundary), bounding circuit only (ignoring its interior), closed (including both its boundary and its interior), and self-intersecting with varying densities of different regions.

In elementary geometry, a polytope is a geometric object with flat sides (faces). It is a generalization in any number of dimensions of the three-dimensional polyhedron. Polytopes may exist in any general number of dimensions n as an n-dimensional polytope or n-polytope. In this context, "flat sides" means that the sides of a (k + 1)-polytope consist of k-polytopes that may have (k – 1)-polytopes in common. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope.

Some theories further generalize the idea to include such objects as unbounded apeirotopes and tessellations, decompositions or tilings of curved manifolds including spherical polyhedra, and set-theoretic abstract polytopes.

Polytopes of more than three dimensions were first discovered by Ludwig Schläfli before 1853, who called such a figure a polyschem. The German term polytop was coined by the mathematician Reinhold Hoppe, and was introduced to English mathematicians as polytope by Alicia Boole Stott.

Approaches to definition

Nowadays, the term polytope is a broad term that covers a wide class of objects, and various definitions appear in the mathematical literature. Many of these definitions are not equivalent to each other, resulting in different overlapping sets of objects being called polytopes. They represent different approaches to generalizing the convex polytopes to include other objects with similar properties.

The original approach broadly followed by Ludwig Schläfli, Thorold Gosset and others begins with the extension by analogy into four or more dimensions, of the idea of a polygon and polyhedron respectively in two and three dimensions.

Attempts to generalise the Euler characteristic of polyhedra to higher-dimensional polytopes led to the development of topology and the treatment of a decomposition or CW-complex as analogous to a polytope. In this approach, a polytope may be regarded as a tessellation or decomposition of some given manifold. An example of this approach defines a polytope as a set of points that admits a simplicial decomposition. In this definition, a polytope is the union of finitely many simplices, with the additional property that, for any two simplices that have a nonempty intersection, their intersection is a vertex, edge, or higher dimensional face of the two. However this definition does not allow star polytopes with interior structures, and so is restricted to certain areas of mathematics.

The discovery of star polyhedra and other unusual constructions led to the idea of a polyhedron as a bounding surface, ignoring its interior. In this light convex polytopes in p-space are equivalent to tilings of the (p−1)-sphere, while others may be tilings of other elliptic, flat or toroidal (p−1)-surfaces – see elliptic tiling and toroidal polyhedron. A polyhedron is understood as a surface whose faces are polygons, a 4-polytope as a hypersurface whose facets (cells) are polyhedra, and so forth.

The idea of constructing a higher polytope from those of lower dimension is also sometimes extended downwards in dimension, with an (edge) seen as a 1-polytope bounded by a point pair, and a point or vertex as a 0-polytope. This approach is used for example in the theory of abstract polytopes.

In certain fields of mathematics, the terms "polytope" and "polyhedron" are used in a different sense: a polyhedron is the generic object in any dimension (referred to as polytope in this article) and polytope means a bounded polyhedron. This terminology is typically confined to polytopes and polyhedra that are convex. With this terminology, a convex polyhedron is the intersection of a finite number of halfspaces and is defined by its sides while a convex polytope is the convex hull of a finite number of points and is defined by its vertices.

Polytopes in lower numbers of dimensions have standard names:

Dimension of polytope	Description
−1	Nullitope
0	Monon
1	Dion
2	Polygon
3	Polyhedron
4	Polychoron

Elements

A polytope comprises elements of different dimensionality such as vertices, edges, faces, cells and so on. Terminology for these is not fully consistent across different authors. For example, some authors use face to refer to an (n − 1)-dimensional element while others use face to denote a 2-face specifically. Authors may use j-face or j-facet to indicate an element of j dimensions. Some use edge to refer to a ridge, while H. S. M. Coxeter uses cell to denote an (n − 1)-dimensional element.

The terms adopted in this article are given in the table below:

Dimension of element	Term (in an n-polytope)
−1	Nullity (necessary in abstract theory)
0	Vertex
1	Edge
2	Face
3	Cell
${\displaystyle \vdots }$	${\displaystyle \vdots }$
j	j-face – element of rank j = −1, 0, 1, 2, 3, ..., n
${\displaystyle \vdots }$	${\displaystyle \vdots }$
n − 3	Peak – (n − 3)-face
n − 2	Ridge or subfacet – (n − 2)-face
n − 1	Facet – (n − 1)-face
n	The polytope itself

An n-dimensional polytope is bounded by a number of (n − 1)-dimensional facets. These facets are themselves polytopes, whose facets are (n − 2)-dimensional ridges of the original polytope. Every ridge arises as the intersection of two facets (but the intersection of two facets need not be a ridge). Ridges are once again polytopes whose facets give rise to (n − 3)-dimensional boundaries of the original polytope, and so on. These bounding sub-polytopes may be referred to as faces, or specifically j-dimensional faces or j-faces. A 0-dimensional face is called a vertex, and consists of a single point. A 1-dimensional face is called an edge, and consists of a line segment. A 2-dimensional face consists of a polygon, and a 3-dimensional face, sometimes called a cell, consists of a polyhedron.

Important classes of polytopes

Convex polytopes

Main article: Convex polytope

A polytope may be convex. The convex polytopes are the simplest kind of polytopes, and form the basis for several different generalizations of the concept of polytopes. A convex polytope is sometimes defined as the intersection of a set of half-spaces. This definition allows a polytope to be neither bounded nor finite. Polytopes are defined in this way, e.g., in linear programming. A polytope is bounded if there is a ball of finite radius that contains it. A polytope is said to be pointed if it contains at least one vertex. Every bounded nonempty polytope is pointed. An example of a non-pointed polytope is the set ${\displaystyle \{(x,y)\in \mathbb {R} ^{2}\mid x\geq 0\}}$. A polytope is finite if it is defined in terms of a finite number of objects, e.g., as an intersection of a finite number of half-planes. It is an integral polytope if all of its vertices have integer coordinates.

A certain class of convex polytopes are reflexive polytopes. An integral ${\displaystyle d}$-polytope ${\displaystyle {\mathcal {P}}}$ is reflexive if for some integral matrix ${\displaystyle \mathbf {A} }$, ${\displaystyle {\mathcal {P}}=\{\mathbf {x} \in \mathbb {R} ^{d}:\mathbf {Ax} \leq \mathbf {1} \}}$, where ${\displaystyle \mathbf {1} }$ denotes a vector of all ones, and the inequality is component-wise. It follows from this definition that ${\displaystyle {\mathcal {P}}}$ is reflexive if and only if ${\displaystyle (t+1){\mathcal {P}}^{\circ }\cap \mathbb {Z} ^{d}=t{\mathcal {P}}\cap \mathbb {Z} ^{d}}$ for all ${\displaystyle t\in \mathbb {Z} _{\geq 0}}$. In other words, a ${\displaystyle (t+1)}$-dilate of ${\displaystyle {\mathcal {P}}}$ differs, in terms of integer lattice points, from a ${\displaystyle t}$-dilate of ${\displaystyle {\mathcal {P}}}$ only by lattice points gained on the boundary. Equivalently, ${\displaystyle {\mathcal {P}}}$ is reflexive if and only if its dual polytope ${\displaystyle {\mathcal {P}}^{*}}$ is an integral polytope.

Regular polytopes

Main article: Regular polytope

Regular polytopes have the highest degree of symmetry of all polytopes. The symmetry group of a regular polytope acts transitively on its flags; hence, the dual polytope of a regular polytope is also regular.

There are three main classes of regular polytope which occur in any number of dimensions:

• Simplices, including the equilateral triangle and the regular tetrahedron.
• Hypercubes or measure polytopes, including the square and the cube.
• Orthoplexes or cross polytopes, including the square and regular octahedron.

Dimensions two, three and four include regular figures which have fivefold symmetries and some of which are non-convex stars, and in two dimensions there are infinitely many regular polygons of n-fold symmetry, both convex and (for n ≥ 5) star. But in higher dimensions there are no other regular polytopes.

In three dimensions the convex Platonic solids include the fivefold-symmetric dodecahedron and icosahedron, and there are also four star Kepler-Poinsot polyhedra with fivefold symmetry, bringing the total to nine regular polyhedra.

In four dimensions the regular 4-polytopes include one additional convex solid with fourfold symmetry and two with fivefold symmetry. There are ten star Schläfli-Hess 4-polytopes, all with fivefold symmetry, giving in all sixteen regular 4-polytopes.

Star polytopes

Main article: Star polytope

A non-convex polytope may be self-intersecting; this class of polytopes include the star polytopes. Some regular polytopes are stars.

Properties

Euler characteristic

Since a (filled) convex polytope P in ${\displaystyle d}$ dimensions is contractible to a point, the Euler characteristic ${\displaystyle \chi }$ of its boundary ∂P is given by the alternating sum:

${\displaystyle \chi =n_{0}-n_{1}+n_{2}-\cdots \pm n_{d-1}=1+(-1)^{d-1}}$, where ${\displaystyle n_{j}}$ is the number of ${\displaystyle j}$-dimensional faces.

This generalizes Euler's formula for polyhedra.

Internal angles

The Gram–Euler theorem similarly generalizes the alternating sum of internal angles ${\textstyle \sum \varphi }$ for convex polyhedra to higher-dimensional polytopes:

${\displaystyle \sum \varphi =(-1)^{d-1}}$

Generalisations of a polytope

Infinite polytopes

Main article: Apeirotope

Not all manifolds are finite. Where a polytope is understood as a tiling or decomposition of a manifold, this idea may be extended to infinite manifolds. plane tilings, space-filling (honeycombs) and hyperbolic tilings are in this sense polytopes, and are sometimes called apeirotopes because they have infinitely many cells.

Among these, there are regular forms including the regular skew polyhedra and the infinite series of tilings represented by the regular apeirogon, square tiling, cubic honeycomb, and so on.

Abstract polytopes

Main article: Abstract polytope

The theory of abstract polytopes attempts to detach polytopes from the space containing them, considering their purely combinatorial properties. This allows the definition of the term to be extended to include objects for which it is difficult to define an intuitive underlying space, such as the 11-cell.

An abstract polytope is a partially ordered set of elements or members, which obeys certain rules. It is a purely algebraic structure, and the theory was developed in order to avoid some of the issues which make it difficult to reconcile the various geometric classes within a consistent mathematical framework. A geometric polytope is said to be a realization in some real space of the associated abstract polytope.

Complex polytopes

Main article: Complex polytope

Structures analogous to polytopes exist in complex Hilbert spaces ${\displaystyle \mathbb {C} ^{n}}$ where n real dimensions are accompanied by n imaginary ones. Regular complex polytopes are more appropriately treated as configurations.

Duality

Every n-polytope has a dual structure, obtained by interchanging its vertices for facets, edges for ridges, and so on generally interchanging its (j − 1)-dimensional elements for (n − j)-dimensional elements (for j = 1 to n − 1), while retaining the connectivity or incidence between elements.

For an abstract polytope, this simply reverses the ordering of the set. This reversal is seen in the Schläfli symbols for regular polytopes, where the symbol for the dual polytope is simply the reverse of the original. For example, {4, 3, 3} is dual to {3, 3, 4}.

In the case of a geometric polytope, some geometric rule for dualising is necessary, see for example the rules described for dual polyhedra. Depending on circumstance, the dual figure may or may not be another geometric polytope.

If the dual is reversed, then the original polytope is recovered. Thus, polytopes exist in dual pairs.

Self-dual polytopes

The 5-cell (4-simplex) is self-dual with 5 vertices and 5 tetrahedral cells.

If a polytope has the same number of vertices as facets, of edges as ridges, and so forth, and the same connectivities, then the dual figure will be similar to the original and the polytope is self-dual.

Some common self-dual polytopes include:

• Every regular n-simplex, in any number of dimensions, with Schläfli symbol {3ⁿ}. These include the equilateral triangle {3}, regular tetrahedron {3,3}, and 5-cell {3,3,3}.
• Every hypercubic honeycomb, in any number of dimensions. These include the apeirogon {∞}, square tiling {4,4} and cubic honeycomb {4,3,4}.
• Numerous compact, paracompact and noncompact hyperbolic tilings, such as the icosahedral honeycomb {3,5,3}, and order-5 pentagonal tiling {5,5}.
• In 2 dimensions, all regular polygons (regular 2-polytopes)
• In 3 dimensions, the canonical polygonal pyramids and elongated pyramids, and tetrahedrally diminished dodecahedron.
• In 4 dimensions, the 24-cell, with Schläfli symbol {3,4,3}. Also the great 120-cell {5,5/2,5} and grand stellated 120-cell {5/2,5,5/2}.

History

Polygons and polyhedra have been known since ancient times.

An early hint of higher dimensions came in 1827 when August Ferdinand Möbius discovered that two mirror-image solids can be superimposed by rotating one of them through a fourth mathematical dimension. By the 1850s, a handful of other mathematicians such as Arthur Cayley and Hermann Grassmann had also considered higher dimensions.

Ludwig Schläfli was the first to consider analogues of polygons and polyhedra in these higher spaces. He described the six convex regular 4-polytopes in 1852 but his work was not published until 1901, six years after his death. By 1854, Bernhard Riemann's Habilitationsschrift had firmly established the geometry of higher dimensions, and thus the concept of n-dimensional polytopes was made acceptable. Schläfli's polytopes were rediscovered many times in the following decades, even during his lifetime.

In 1882 Reinhold Hoppe, writing in German, coined the word polytop to refer to this more general concept of polygons and polyhedra. In due course Alicia Boole Stott, daughter of logician George Boole, introduced the anglicised polytope into the English language.

In 1895, Thorold Gosset not only rediscovered Schläfli's regular polytopes but also investigated the ideas of semiregular polytopes and space-filling tessellations in higher dimensions. Polytopes also began to be studied in non-Euclidean spaces such as hyperbolic space.

An important milestone was reached in 1948 with H. S. M. Coxeter's book Regular Polytopes, summarizing work to date and adding new findings of his own.

Meanwhile, the French mathematician Henri Poincaré had developed the topological idea of a polytope as the piecewise decomposition (e.g. CW-complex) of a manifold. Branko Grünbaum published his influential work on Convex Polytopes in 1967.

In 1952 Geoffrey Colin Shephard generalised the idea as complex polytopes in complex space, where each real dimension has an imaginary one associated with it. Coxeter developed the theory further.

The conceptual issues raised by complex polytopes, non-convexity, duality and other phenomena led Grünbaum and others to the more general study of abstract combinatorial properties relating vertices, edges, faces and so on. A related idea was that of incidence complexes, which studied the incidence or connection of the various elements with one another. These developments led eventually to the theory of abstract polytopes as partially ordered sets, or posets, of such elements. Peter McMullen and Egon Schulte published their book Abstract Regular Polytopes in 2002.

Enumerating the uniform polytopes, convex and nonconvex, in four or more dimensions remains an outstanding problem. The convex uniform 4-polytopes were fully enumerated by John Conway and Michael Guy using a computer in 1965; in higher dimensions this problem was still open as of 1997. The full enumeration for nonconvex uniform polytopes is not known in dimensions four and higher as of 2008.

In modern times, polytopes and related concepts have found many important applications in fields as diverse as computer graphics, optimization, search engines, cosmology, quantum mechanics and numerous other fields. In 2013 the amplituhedron was discovered as a simplifying construct in certain calculations of theoretical physics.

Applications

In the field of optimization, linear programming studies the maxima and minima of linear functions; these maxima and minima occur on the boundary of an n-dimensional polytope. In linear programming, polytopes occur in the use of generalized barycentric coordinates and slack variables.

In twistor theory, a branch of theoretical physics, a polytope called the amplituhedron is used in to calculate the scattering amplitudes of subatomic particles when they collide. The construct is purely theoretical with no known physical manifestation, but is said to greatly simplify certain calculations.

3D modeling

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/3D_modeling 

Three-dimensional (3D) computer graphics

In 3D computer graphics, 3D modeling is the process of developing a mathematical coordinate-based representation of any surface of an object (inanimate or living) in three dimensions via specialized software by manipulating edges, vertices, and polygons in a simulated 3D space.

Three-dimensional (3D) models represent a physical body using a collection of points in 3D space, connected by various geometric entities such as triangles, lines, curved surfaces, etc. Being a collection of data (points and other information), 3D models can be created manually, algorithmically (procedural modeling), or by scanning. Their surfaces may be further defined with texture mapping.

Outline

See also: Environment artist

The product is called a 3D model. Someone who works with 3D models may be referred to as a 3D artist or a 3D modeler.

A 3D Model can also be displayed as a two-dimensional image through a process called 3D rendering or used in a computer simulation of physical phenomena.

3D Models may be created automatically or manually. The manual modeling process of preparing geometric data for 3D computer graphics is similar to plastic arts such as sculpting. The 3D model can be physically created using 3D printing devices that form 2D layers of the model with three-dimensional material, one layer at a time. Without a 3D model, a 3D print is not possible.

3D modeling software is a class of 3D computer graphics software used to produce 3D models. Individual programs of this class, such as SketchUp, are called modeling applications.

History

Three-dimensional model of a spectrograph

 

Rotating 3D video-game model

3D selfie models are generated from 2D pictures taken at the Fantasitron 3D photo booth at Madurodam

3D models are now widely used anywhere in 3D graphics and CAD but their history predates the widespread use of 3D graphics on personal computers.

In the past, many computer games used pre-rendered images of 3D models as sprites before computers could render them in real-time. The designer can then see the model in various directions and views, this can help the designer see if the object is created as intended to compared to their original vision. Seeing the design this way can help the designer or company figure out changes or improvements needed to the product.

Representation

A modern render of the iconic Utah teapot model developed by Martin Newell (1975). The Utah teapot is one of the most common models used in 3D graphics education.

Almost all 3D models can be divided into two categories:

• Solid – These models define the volume of the object they represent (like a rock). Solid models are mostly used for engineering and medical simulations, and are usually built with constructive solid geometry
• Shell or boundary – These models represent the surface, i.e. the boundary of the object, not its volume (like an infinitesimally thin eggshell). Almost all visual models used in games and film are shell models.

Solid and shell modeling can create functionally identical objects. Differences between them are mostly variations in the way they are created and edited and conventions of use in various fields and differences in types of approximations between the model and reality.

Shell models must be manifold (having no holes or cracks in the shell) to be meaningful as a real object. In a shell model of a cube, the bottom and top surface of the cube must have a uniform thickness with no holes or cracks in the first and last layer printed. Polygonal meshes (and to a lesser extent subdivision surfaces) are by far the most common representation. Level sets are a useful representation for deforming surfaces which undergo many topological changes such as fluids.

The process of transforming representations of objects, such as the middle point coordinate of a sphere and a point on its circumference into a polygon representation of a sphere, is called tessellation. This step is used in polygon-based rendering, where objects are broken down from abstract representations ("primitives") such as spheres, cones etc., to so-called meshes, which are nets of interconnected triangles. Meshes of triangles (instead of e.g. squares) are popular as they have proven to be easy to rasterize (the surface described by each triangle is planar, so the projection is always convex). Polygon representations are not used in all rendering techniques, and in these cases the tessellation step is not included in the transition from abstract representation to rendered scene.

Process

There are three popular ways to represent a model:

• Polygonal modeling – Points in 3D space, called vertices, are connected by line segments to form a polygon mesh. The vast majority of 3D models today are built as textured polygonal models, because they are flexible, because computers can render them so quickly. However, polygons are planar and can only approximate curved surfaces using many polygons.
• Curve modeling – Surfaces are defined by curves, which are influenced by weighted control points. The curve follows (but does not necessarily interpolate) the points. Increasing the weight for a point will pull the curve closer to that point. Curve types include nonuniform rational B-spline (NURBS), splines, patches, and geometric primitives
• Digital sculpting – Still a fairly new method of modeling, 3D sculpting has become very popular in the few years it has been around. There are currently three types of digital sculpting: Displacement, which is the most widely used among applications at this moment, uses a dense model (often generated by subdivision surfaces of a polygon control mesh) and stores new locations for the vertex positions through use of an image map that stores the adjusted locations. Volumetric, loosely based on voxels, has similar capabilities as displacement but does not suffer from polygon stretching when there are not enough polygons in a region to achieve a deformation. Dynamic tessellation, which is similar to voxel, divides the surface using triangulation to maintain a smooth surface and allow finer details. These methods allow for very artistic exploration as the model will have a new topology created over it once the models form and possibly details have been sculpted. The new mesh will usually have the original high resolution mesh information transferred into displacement data or normal map data if for a game engine.

A 3D fantasy fish composed of organic surfaces generated using LAI4D.

The modeling stage consists of shaping individual objects that are later used in the scene. There are a number of modeling techniques, including:

• Constructive solid geometry
• Implicit surfaces
• Subdivision surfaces

Modeling can be performed by means of a dedicated program (e.g., Blender, Cinema 4D, LightWave, Maya, Modo, 3ds Max) or an application component (Shaper, Lofter in 3ds Max) or some scene description language (as in POV-Ray). In some cases, there is no strict distinction between these phases; in such cases modeling is just part of the scene creation process (this is the case, for example, with Caligari trueSpace and Realsoft 3D).

3D models can also be created using the technique of Photogrammetry with dedicated programs such as RealityCapture, Metashape and 3DF Zephyr. Cleanup and further processing can be performed with applications such as MeshLab, the GigaMesh Software Framework, netfabb or MeshMixer. Photogrammetry creates models using algorithms to interpret the shape and texture of real-world objects and environments based on photographs taken from many angles of the subject.

Complex materials such as blowing sand, clouds, and liquid sprays are modeled with particle systems, and are a mass of 3D coordinates which have either points, polygons, texture splats, or sprites assigned to them.

Human models

Main article: Virtual actor

The first widely available commercial application of human virtual models appeared in 1998 on the Lands' End web site. The human virtual models were created by the company My Virtual Mode Inc. and enabled users to create a model of themselves and try on 3D clothing. There are several modern programs that allow for the creation of virtual human models (Poser being one example).

3D clothing

Dynamic 3D clothing model made in Marvelous Designer

The development of cloth simulation software such as Marvelous Designer, CLO3D and Optitex, has enabled artists and fashion designers to model dynamic 3D clothing on the computer. Dynamic 3D clothing is used for virtual fashion catalogs, as well as for dressing 3D characters for video games, 3D animation movies, for digital doubles in movies as well as for making clothes for avatars in virtual worlds such as SecondLife.

Comparison with 2D methods

3D photorealistic effects are often achieved without wire-frame modeling and are sometimes indistinguishable in the final form. Some graphic art software includes filters that can be applied to 2D vector graphics or 2D raster graphics on transparent layers.

Advantages of wireframe 3D modeling over exclusively 2D methods include:

• Flexibility, ability to change angles or animate images with quicker rendering of the changes;
• Ease of rendering, automatic calculation and rendering photorealistic effects rather than mentally visualizing or estimating;
• Accurate photorealism, less chance of human error in misplacing, overdoing, or forgetting to include a visual effect.

Disadvantages compare to 2D photorealistic rendering may include a software learning curve and difficulty achieving certain photorealistic effects. Some photorealistic effects may be achieved with special rendering filters included in the 3D modeling software. For the best of both worlds, some artists use a combination of 3D modeling followed by editing the 2D computer-rendered images from the 3D model.

3D model market

A large market for 3D models (as well as 3D-related content, such as textures, scripts, etc.) still exists – either for individual models or large collections. Several online marketplaces for 3D content allow individual artists to sell content that they have created, including TurboSquid, CGStudio, CreativeMarket, MyMiniFactory, Sketchfab, CGTrader and Cults. Often, the artists' goal is to get additional value out of assets they have previously created for projects. By doing so, artists can earn more money out of their old content, and companies can save money by buying pre-made models instead of paying an employee to create one from scratch. These marketplaces typically split the sale between themselves and the artist that created the asset, artists get 40% to 95% of the sales according to the marketplace. In most cases, the artist retains ownership of the 3d model while the customer only buys the right to use and present the model. Some artists sell their products directly in its own stores offering their products at a lower price by not using intermediaries.

Over the last several years numerous marketplaces specializing in 3D rendering and printing models have emerged. Some of the 3D printing marketplaces are a combination of models sharing sites, with or without a built in e-com capability. Some of those platforms also offer 3D printing services on demand, software for model rendering and dynamic viewing of items. 3D printing file sharing and model rendering platforms include Shapeways, Sketchfab, Pinshape, Thingiverse, TurboSquid, CGTrader, Threeding, MyMiniFactory, and GrabCAD.

3D printing

Main articles: 3D printing and Rapid prototyping

The term 3D printing or three-dimensional printing is a form of additive manufacturing technology where a three-dimensional object is created from successive layers material. Objects can be created without the need for complex expensive molds or assembly with multiple parts. 3D printing allows ideas to be prototyped and tested without having to go through a production process.

In recent years, there has been an upsurge in the number of companies offering personalized 3D printed models of objects that have been scanned, designed in CAD software, and then printed to the customer's requirements. 3D models can be purchased from online marketplaces and printed by individuals or companies using commercially available 3D printers, enabling the home-production of objects such as spare parts and even medical equipment.

Uses

Steps of forensic facial reconstruction of a mummy made in Blender by the Brazilian 3D designer Cícero Moraes.

Today, 3D modeling is used in various industries like film, animation and gaming, interior design and architecture. They are also used in the medical industry to create interactive representations of anatomy.

The medical industry uses detailed models of organs; these may be created with multiple 2-D image slices from an MRI or CT scan. The movie industry uses them as characters and objects for animated and real-life motion pictures. The video game industry uses them as assets for computer and video games.

The science sector uses them as highly detailed models of chemical compounds.

The architecture industry uses them to demonstrate proposed buildings and landscapes in lieu of traditional, physical architectural models.

The archaeology community is now creating 3D models of cultural heritage for research and visualization.

The engineering community utilizes them as designs of new devices, vehicles and structures as well as a host of other uses.

In recent decades the earth science community has started to construct 3D geological models as a standard practice.

3D models can also be the basis for physical devices that are built with 3D printers or CNC machines.

In terms of video game development, 3D modeling is one stage in a longer development process. Simply put, the source of the geometry for the shape of an object can be:

1. A designer, industrial engineer or artist using a 3D-CAD system
2. An existing object, reverse engineered or copied using a 3-D shape digitizer or scanner
3. Mathematical data stored in memory based on a numerical description or calculation of the object.

A wide number of 3D software are also used in constructing digital representation of mechanical models or parts before they are actually manufactured. CAD- and CAM-related software is used in such fields, and with this software, not only can you construct the parts, but also assemble them, and observe their functionality.

3D modeling is also used in the field of industrial design, wherein products are 3D modeled before representing them to the clients. In media and event industries, 3D modeling is used in stage and set design.

The OWL 2 translation of the vocabulary of X3D can be used to provide semantic descriptions for 3D models, which is suitable for indexing and retrieval of 3D models by features such as geometry, dimensions, material, texture, diffuse reflection, transmission spectra, transparency, reflectivity, opalescence, glazes, varnishes, and enamels (as opposed to unstructured textual descriptions or 2.5D virtual museums and exhibitions using Google Street View on Google Arts & Culture, for example). The RDF representation of 3D models can be used in reasoning, which enables intelligent 3D applications which, for example, can automatically compare two 3D models by volume.

Testing a 3D solid model

Further information: Solid modeling

3D solid models can be tested in different ways depending on what is needed by using simulation, mechanism design, and analysis. If a motor is designed and assembled correctly (this can be done differently depending on what 3D modeling program is being used), using the mechanism tool the user should be able to tell if the motor or machine is assembled correctly by how it operates. Different design will need to be tested in different ways. For example; a pool pump would need a simulation ran of the water running through the pump to see how the water flows through the pump. These tests verify if a product is developed correctly or if it needs to be modified to meet its requirements.

Conformational isomerism

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Conformational_isomerism 

 

Rotation about single bond of butane to interconvert one conformation to another. The gauche conformation on the right is a conformer, while the eclipsed conformation on the left is a transition state between conformers. Above: Newman projection; below: depiction of spatial orientation.

In chemistry, conformational isomerism is a form of stereoisomerism in which the isomers can be interconverted just by rotations about formally single bonds (refer to figure on single bond rotation). While any two arrangements of atoms in a molecule that differ by rotation about single bonds can be referred to as different conformations, conformations that correspond to local minima on the potential energy surface are specifically called conformational isomers or conformers. Conformations that correspond to local maxima on the energy surface are the transition states between the local-minimum conformational isomers. Rotations about single bonds involve overcoming a rotational energy barrier to interconvert one conformer to another. If the energy barrier is low, there is free rotation and a sample of the compound exists as a rapidly equilibrating mixture of multiple conformers; if the energy barrier is high enough then there is restricted rotation, a molecule may exist for a relatively long time period as a stable rotational isomer or rotamer (an isomer arising from hindered single-bond rotation). When the time scale for interconversion is long enough for isolation of individual rotamers (usually arbitrarily defined as a half-life of interconversion of 1000 seconds or longer), the isomers are termed atropisomers (see: atropisomerism). The ring-flip of substituted cyclohexanes constitutes another common form of conformational isomerism.

Conformational isomers are thus distinct from the other classes of stereoisomers (i. e. configurational isomers) where interconversion necessarily involves breaking and reforming of chemical bonds. For example, L/D- and R/S- configurations of organic molecules have different handedness and optical activities, and can only be interconverted by breaking one or more bonds connected to the chiral atom and reforming a similar bond in a different direction or spatial orientation. They also differ from geometric (cis/trans) isomers, another class of stereoisomers, which require the π-component of double bonds to break for interconversion. (Although the distinction is not always clear-cut, since certain bonds that are formally single bonds actually have double bond character that becomes apparent only when secondary resonance contributors are considered, like the C–N bonds of amides, for instance.) Due to rapid interconversion, conformers are usually not isolable at room temperature.

The study of the energetics between different conformations is referred to as conformational analysis. It is useful for understanding the stability of different isomers, for example, by taking into account the spatial orientation and through-space interactions of substituents. In addition, conformational analysis can be used to predict and explain product selectivity, mechanisms, and rates of reactions. Conformational analysis also plays an important role in rational, structure-based drug design.

Types

Relative conformation energy diagram of butane as a function of dihedral angle. A: antiperiplanar, anti or trans. B: synclinal or gauche. C: anticlinal or eclipsed. D: synperiplanar or cis.

Rotating their carbon–carbon bonds, the molecules ethane and propane have three local energy minima. They are structurally and energetically equivalent, and are called the staggered conformers. For each molecule, the three substituents emanating from each carbon–carbon bond are staggered, with each H–C–C–H dihedral angle (and H–C–C–CH₃ dihedral angle in the case of propane) equal to 60° (or approximately equal to 60° in the case of propane). The three eclipsed conformations, in which the dihedral angles are zero, are transition states (energy maxima) connecting two equivalent energy minima, the staggered conformers.

The butane molecule is the simplest molecule for which single bond rotations result in two types of nonequivalent structures, known as the anti- and gauche-conformers (see figure).

For example, butane has three conformers relating to its two methyl (CH₃) groups: two gauche conformers, which have the methyls ±60° apart and are enantiomeric, and an anti conformer, where the four carbon centres are coplanar and the substituents are 180° apart (refer to free energy diagram of butane). The energy difference between gauche and anti is 0.9 kcal/mol associated with the strain energy of the gauche conformer. The anti conformer is, therefore, the most stable (≈ 0 kcal/mol). The three eclipsed conformations with dihedral angles of 0°, 120°, and 240° are transition states between conformers. Note that the two eclipsed conformations have different energies: at 0° the two methyl groups are eclipsed, resulting in higher energy (≈ 5 kcal/mol) than at 120°, where the methyl groups are eclipsed with hydrogens (≈ 3.5 kcal/mol).

While simple molecules can be described by these types of conformations, more complex molecules require the use of the Klyne–Prelog system to describe the different conformers.

More specific examples of conformational isomerism are detailed elsewhere:

• Ring conformation
  • Cyclohexane conformations, including with chair and boat conformations among others.
  • Cycloalkane conformations, including medium rings and macrocycles
  • Carbohydrate conformation, which includes cyclohexane conformations as well as other details.
• Allylic strain – energetics related to rotation about the single bond between an sp² carbon and an sp³ carbon.
• Atropisomerism – due to restricted rotation about a bond.
• Folding, including the secondary and tertiary structure of biopolymers (nucleic acids and proteins).
• Akamptisomerism – due to restricted inversion of a bond angle.

Free energy and equilibria of conformational isomers

Equilibrium of conformers

Equilibrium distribution of two conformers at different temperatures given the free energy of their interconversion.

Conformational isomers exist in a dynamic equilibrium, where the relative free energies of isomers determines the population of each isomer and the energy barrier of rotation determines the rate of interconversion between isomers:

${\displaystyle K=e^{-\Delta G^{\circ }/RT},}$

where K is the equilibrium constant, ΔG° is the difference in standard free energy between the two conformers in kcal/mol, R is the universal gas constant (1.987×10⁻³ kcal/mol K), and T is the system's temperature in kelvins. In units of kcal/mol at 298 K,

${\displaystyle K\approx 10^{-\Delta G^{\circ }/(1.36{\text{ kcal}}/\mathrm {mol} )}.}$

Thus, every 1.36 kcal/mol corresponds to a factor of about 10 in term of equilibrium constant at temperatures around room temperature. (The "1.36 rule" is useful in general for estimation of equilibrium constants at room temperature from free energy differences. At lower temperatures, a smaller energy difference is needed to obtain a given equilibrium constant.)

Three isotherms are given in the diagram depicting the equilibrium distribution of two conformers at different temperatures. At a free energy difference of 0 kcal/mol, this gives an equilibrium constant of 1, meaning that two conformers exist in a 1:1 ratio. The two have equal free energy; neither is more stable, so neither predominates compared to the other. A negative difference in free energy means that a conformer interconverts to a thermodynamically more stable conformation, thus the equilibrium constant will always be greater than 1. For example, the ΔG° for the transformation of butane from the gauche conformer to the anti conformer is −0.47 kcal/mol at 298 K. This gives an equilibrium constant is about 2.2 in favor of the anti conformer, or a 31:69 mixture of gauche:anti conformers at equilibrium. Conversely, a positive difference in free energy means the conformer already is the more stable one, so the interconversion is an unfavorable equilibrium (K < 1). Even for highly unfavorable changes (large positive ΔG°), the equilibrium constant between two conformers can be increased by increasing the temperature, so that the amount of the less stable conformer present at equilibrium increases (although it always remains the minor conformer).

Population distribution of conformers

Boltzmann distribution % of lowest energy conformation in a two component equilibrating system at various temperatures (°C, color) and energy difference in kcal/mol (x-axis)

The fractional population distribution of different conformers follows a Boltzmann distribution:

${\displaystyle {\frac {N_{i}}{N_{\text{total}}}}={\frac {e^{-E_{i}/RT}}{\sum _{k=1}^{M}e^{-E_{k}/RT}}}.}$

The left hand side is the proportion of conformer i in an equilibrating mixture of M conformers in thermodynamic equilibrium. On the right side, E_k (k = 1, 2, ..., M) is the energy of conformer k, R is the molar ideal gas constant (approximately equal to 8.314 J/(mol·K) or 1.987 cal/(mol·K)), and T is the absolute temperature. The denominator of the right side is the partition function.

Factors contributing to the free energy of conformers

The effects of electrostatic and steric interactions of the substituents as well as orbital interactions such as hyperconjugation are responsible for the relative stability of conformers and their transition states. The contributions of these factors vary depending on the nature of the substituents and may either contribute positively or negatively to the energy barrier. Computational studies of small molecules such as ethane suggest that electrostatic effects make the greatest contribution to the energy barrier; however, the barrier is traditionally attributed primarily to steric interactions.

Contributions to rotational energy barrier

In the case of cyclic systems, the steric effect and contribution to the free energy can be approximated by A values, which measure the energy difference when a substituent on cyclohexane in the axial as compared to the equatorial position. In large (>14 atom) rings, there are many accessible low-energy conformations which correspond to the strain-free diamond lattice.

Isolation or observation of the conformational isomers

The short timescale of interconversion precludes the separation of conformational isomers in most cases. Atropisomers are conformational isomers which can be separated due to restricted rotation. The equilibrium between conformational isomers can be observed using a variety of spectroscopic techniques.

Protein folding also generates stable conformational isomers which can be observed. The Karplus equation relates the dihedral angle of vicinal protons to their J-coupling constants as measured by NMR. The equation aids in the elucidation of protein folding as well as the conformations of other rigid aliphatic molecules. Protein side chains exhibit rotamers, whose distribution is determined by their steric interaction with different conformations of the backbone. This is evident from statistical analysis of the conformations of protein side chains in the Backbone-dependent rotamer library.

In cyclohexane derivatives, the two chair conformers interconvert with rapidly at room temperature, with cyclohexane itself undergoing the ring-flip at a rates of approximately 10⁵ ring-flips/sec, with an overall energy barrier of 10 kcal/mol (42 kJ/mol), which precludes their separation at ambient temperatures. However, at low temperatures below the coalescence point one can directly monitor the equilibrium by NMR spectroscopy and by dynamic, temperature dependent NMR spectroscopy the barrier interconversion.

The dynamics of conformational (and other kinds of) isomerism can be monitored by NMR spectroscopy at varying temperatures. The technique applies to barriers of 8–14 kcal/mol, and species exhibiting such dynamics are often called "fluxional".

Besides NMR spectroscopy, IR spectroscopy is used to measure conformer ratios. For the axial and equatorial conformer of bromocyclohexane, ν_CBr differs by almost 50 cm⁻¹.

Conformation-dependent reactions

Reaction rates are highly dependent on the conformation of the reactants. In many cases the dominant product arises from the reaction of the less prevalent conformer, by virtue of the Curtin-Hammett principle. This is typical for situations where the conformational equilibration is much faster than reaction to form the product. The dependence of a reaction on the stereochemical orientation is therefore usually only visible in configurational isomers, in which a particular conformation is locked by substituents. Prediction of rates of many reactions involving the transition between sp2 and sp3 states, such as ketone reduction, alcohol oxidation or nucleophilic substitution is possible if all conformers and their relative stability ruled by their strain is taken into account.

One example with configurational isomers is provided by elimination reactions, which involve the simultaneous removal of a proton and a leaving group from vicinal or antiperiplanar positions under the influence of a base.

Base-induced bimolecular dehydrohalogenation (an E2 type reaction mechanism). The optimum geometry for the transition state requires the breaking bonds to be antiperiplanar, as they are in the appropriate staggered conformation

The mechanism requires that the departing atoms or groups follow antiparallel trajectories. For open chain substrates this geometric prerequisite is met by at least one of the three staggered conformers. For some cyclic substrates such as cyclohexane, however, an antiparallel arrangement may not be attainable depending on the substituents which might set a conformational lock. Adjacent substituents on a cyclohexane ring can achieve antiperiplanarity only when they occupy trans diaxial positions (that is, both are in axial position, one going up and one going down).

One consequence of this analysis is that trans-4-tert-butylcyclohexyl chloride cannot easily eliminate but instead undergoes substitution (see diagram below) because the most stable conformation has the bulky t-Bu group in the equatorial position, therefore the chloride group is not antiperiplanar with any vicinal hydrogen (it is gauche to all four). The thermodynamically unfavored conformation has the t-Bu group in the axial position, which is higher in energy by more than 5 kcal/mol (see A value). As a result, the t-Bu group "locks" the ring in the conformation where it is in the equatorial position and substitution reaction is observed. On the other hand, cis-4-tert-butylcyclohexyl chloride undergoes elimination because antiperiplanarity of Cl and H can be achieved when the t-Bu group is in the favorable equatorial position.

Thermodynamically unfavored conformation of trans-4-tert-butylcyclohexyl chloride where the t-Bu group is in the axial position exerting 7-atom interactions.

 

The trans isomer can attain antiperiplanarity only via the unfavored axial conformer; therefore, it does not eliminate. The cis isomer is already in the correct geometry in its most stable conformation; therefore, it eliminates easily.

The repulsion between an axial t-butyl group and hydrogen atoms in the 1,3-diaxial position is so strong that the cyclohexane ring will revert to a twisted boat conformation. The strain in cyclic structures is usually characterized by deviations from ideal bond angles (Baeyer strain), ideal torsional angles (Pitzer strain) or transannular (Prelog ) interactions.

Alkane stereochemistry

Alkane conformers arise from rotation around sp³ hybridised carbon–carbon sigma bonds. The smallest alkane with such a chemical bond, ethane, exists as an infinite number of conformations with respect to rotation around the C–C bond. Two of these are recognised as energy minimum (staggered conformation) and energy maximum (eclipsed conformation) forms. The existence of specific conformations is due to hindered rotation around sigma bonds, although a role for hyperconjugation is proposed by a competing theory.

The importance of energy minima and energy maxima is seen by extension of these concepts to more complex molecules for which stable conformations may be predicted as minimum-energy forms. The determination of stable conformations has also played a large role in the establishment of the concept of asymmetric induction and the ability to predict the stereochemistry of reactions controlled by steric effects.

In the example of staggered ethane in Newman projection, a hydrogen atom on one carbon atom has a 60° torsional angle or torsion angle  with respect to the nearest hydrogen atom on the other carbon so that steric hindrance is minimised. The staggered conformation is more stable by 12.5 kJ/mol than the eclipsed conformation, which is the energy maximum for ethane. In the eclipsed conformation the torsional angle is minimised.

In butane, the two staggered conformations are no longer equivalent and represent two distinct conformers:the anti-conformation (left-most, below) and the gauche conformation (right-most, below).

Both conformations are free of torsional strain, but, in the gauche conformation, the two methyl groups are in closer proximity than the sum of their van der Waals radii. The interaction between the two methyl groups is repulsive (van der Waals strain), and an energy barrier results.

A measure of the potential energy stored in butane conformers with greater steric hindrance than the 'anti'-conformer ground state is given by these values:

• Gauche, conformer – 3.8 kJ/mol
• Eclipsed H and CH₃ – 16 kJ/mol
• Eclipsed CH₃ and CH₃ – 19 kJ/mol.

The eclipsed methyl groups exert a greater steric strain because of their greater electron density compared to lone hydrogen atoms.

Relative energies of conformations of butane with respect to rotation of the central C-C bond.

The textbook explanation for the existence of the energy maximum for an eclipsed conformation in ethane is steric hindrance, but, with a C-C bond length of 154 pm and a Van der Waals radius for hydrogen of 120 pm, the hydrogen atoms in ethane are never in each other's way. The question of whether steric hindrance is responsible for the eclipsed energy maximum is a topic of debate to this day. One alternative to the steric hindrance explanation is based on hyperconjugation as analyzed within the Natural Bond Orbital framework. In the staggered conformation, one C-H sigma bonding orbital donates electron density to the antibonding orbital of the other C-H bond. The energetic stabilization of this effect is maximized when the two orbitals have maximal overlap, occurring in the staggered conformation. There is no overlap in the eclipsed conformation, leading to a disfavored energy maximum. On the other hand, an analysis within quantitative molecular orbital theory shows that 2-orbital-4-electron (steric) repulsions are dominant over hyperconjugation. A valence bond theory study also emphasizes the importance of steric effects.

Nomenclature

Naming alkanes per standards listed in the IUPAC Gold Book is done according to the Klyne–Prelog system for specifying angles (called either torsional or dihedral angles) between substituents around a single bond:

• a torsion angle between 0° and ± 90° is called syn (s)
• a torsion angle between ± 90° and 180° is called anti (a)
• a torsion angle between 30° and 150° or between –30° and –150° is called clinal (c)
• a torsion angle between 0° and ± 30° or ± 150° and 180° is called periplanar (p)
• a torsion angle between 0° and ± 30° is called synperiplanar (sp), also called syn- or cis- conformation
• a torsion angle between 30° to 90° and –30° to –90° is called synclinal (sc), also called gauche or skew
• a torsion angle between 90° and 150° or –90° and –150° is called anticlinal (ac)
• a torsion angle between ± 150° and 180° is called antiperiplanar (ap), also called anti- or trans- conformation

Torsional strain or "Pitzer strain" refers to resistance to twisting about a bond.

Special cases

In n-pentane, the terminal methyl groups experience additional pentane interference.

Replacing hydrogen by fluorine in polytetrafluoroethylene changes the stereochemistry from the zigzag geometry to that of a helix due to electrostatic repulsion of the fluorine atoms in the 1,3 positions. Evidence for the helix structure in the crystalline state is derived from X-ray crystallography and from NMR spectroscopy and circular dichroism in solution.