TheInfoList

In
3D computer graphics 3D computer graphics, sometimes called CGI, 3DCG or three-dimensional computer graphics (in contrast to 2D computer graphics 2D computer graphics is the Computer-generated imagery, computer-based generation of digital images—mostly from t ...
, a voxel represents a value on a
regular grid A regular grid is a tessellation A tiling or tessellation of a flat surface is the covering of a plane (mathematics), plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellations can be ...

in
three-dimensional space Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameters) are required to determine the position of an element (i.e., point). This is the informal meaning of the ...
. As with
pixel In digital imaging Digital imaging or digital image acquisition is the creation of a representation of the visual characteristics of an object, such as a physical scene or the interior structure of an object. The term is often assumed to imp ...

s in a 2D
bitmap In computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes and development of both computer hardware , hardware and softwar ...

, voxels themselves do not typically have their position (i.e.
coordinate In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

s) explicitly encoded with their values. Instead, rendering systems infer the position of a voxel based upon its position relative to other voxels (i.e., its position in the
data structure In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of ...

that makes up a single volumetric image). In contrast to pixels and voxels,
polygons In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...
are often explicitly represented by the coordinates of their vertices (as
points Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Scotland, Lismore, Inner Hebrides, ...
). A direct consequence of this difference is that polygons can efficiently represent simple 3D structures with much empty or homogeneously filled space, while voxels excel at representing regularly sampled spaces that are non-homogeneously filled. Voxels are frequently used in the
visualization Visualization or visualisation may refer to: *Visualization (graphics) Visualization or visualisation (see spelling differences Despite the various English dialects spoken from country to country and within different regions of the sam ...
and analysis of
medical Medicine is the science and Praxis (process) , practice of caring for a patient, managing the diagnosis, prognosis, Preventive medicine, prevention, therapy, treatment, Palliative care , palliation of their injury or disease, and Health promoti ...
and scientific data (e.g.
geographic information system A geographic information system (GIS) is a type of database In , a database is an organized collection of stored and accessed electronically from a . Where databases are more complex they are often developed using formal techniques. The ( ...
s (GIS)). Some volumetric displays use voxels to describe their
resolution Resolution(s) may refer to: Common meanings * Resolution (debate), the statement which is debated in policy debate * Resolution (law), a written motion adopted by a deliberative body * New Year's resolution, a commitment that an individual make ...
. For example, a cubic volumetric display might be able to show 512×512×512 (or about 134 million) voxels. One of the definitions is: "Voxel is an image of a three-dimensional space region limited by given sizes, which has its own ''nodal point'' coordinates in an accepted coordinate system, its own ''form'', its own ''state parameter'' that indicates its belonging to some modeled object, and has ''properties'' of modeled region." This definition has the following advantage. If fixed voxel form is used within the whole model it is much easier to operate with voxel ''nodal points'', i.e. three coordinates of this point. Yet, there is the simple form of record – indexes of the elements in the model set, i.e. integer coordinates. Model set elements in this case are ''state parameters,'' indicating voxel belonging to the modeled object or its separate parts, including their surfaces.

# Etymology

The word ''voxel'' originated analogously to the word "
pixel In digital imaging Digital imaging or digital image acquisition is the creation of a representation of the visual characteristics of an object, such as a physical scene or the interior structure of an object. The term is often assumed to imp ...
", with ''vo'' representing "
volume Volume is a scalar quantity expressing the amount Quantity or amount is a property that can exist as a multitude Multitude is a term for a group of people who cannot be classed under any other distinct category, except for their shared fact ...

" (instead of pixel's "picture") and ''el'' representing "element"; a similar formation with ''el'' for "element" is the word "
texel Texel (; Texels dialect: ) is a and an with a population of 13,643 in , . It is the largest and most populated island of the in the . The island is situated north of , northeast of , and southwest of . Name The name ''Texel'' is Frisian, ...
".

# Rendering

A volume described as voxels can be visualized either by direct
volume rendering scan of a forearm with different color schemes for muscle, fat, bone, and blood In scientific visualization Surface rendering of '' confocal_microscope..html" ;"title="onfocal_microscopy.html" ;"title="pollen.html" ;"title="Arabidopsis thal ...
or by the extraction of
polygon In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...

iso-surfaces that follow the contours of given threshold values. The
marching cubes slices using marching-cubes (about 150,000 triangles) Marching cubes is a computer graphics algorithm of an algorithm (Euclid's algorithm) for calculating the greatest common divisor (g.c.d.) of two numbers ''a'' and ''b'' in locations named A ...
algorithm is often used for isosurface extraction, however other methods exist as well. Both ray-tracing and
ray-castingRay casting is the methodological basis for 3-D CAD/CAM solid modeling and image rendering. It is essentially the same as ray tracing (graphics), ray tracing for computer graphics where virtual light rays are "cast" or "traced" on their path from the ...
, as well as
rasterisation Rasterization (or rasterisation) is the task of taking an image described in a vector graphics Vector graphics, as a form of computer graphics, is the set of mechanisms for creating visual images directly from Geometric primitive , geometr ...
, can be applied to voxel data to obtain 2D
raster graphics upright=1, The Smiley, smiley face in the top left corner is a raster image. When enlarged, individual pixels appear as squares. Enlarging further, each pixel can be analyzed, with their colors constructed through combination of the values for r ...
to depict on a monitor.

## Incremental error rasterisation

Another technique for voxels involves raster graphics where one simply raytraces every pixel of the display into the scene, tracking an error term to determine when to step. A typical implementation will raytrace each pixel of the display starting at the bottom of the screen using what is known as a y-buffer. When a voxel is reached that has a higher y value on the display it is added to the y-buffer overriding the previous value and connected with the previous y-value on the screen interpolating the color values. There is a major downside to voxel rasterization when transformation is applied which causes severe
aliasing In signal processing Signal processing is an electrical engineering subfield that focuses on analysing, modifying, and synthesizing signals such as audio signal processing, sound, image processing, images, and scientific measurements. Sig ...

. The advantage was the ability to rasterise using cheap integer calculations on a
CPU A central processing unit (CPU), also called a central processor, main processor or just processor, is the electronic circuit File:PExdcr01CJC.jpg, 200px, A circuit built on a printed circuit board (PCB). An electronic circuit is composed of ...

without
hardware acceleration Hardware acceleration is the use of computer hardware Computer hardware includes the physical parts of a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. ...
. '' Outcast'', and other 1990s video games employed this graphics technique for effects such as
reflectionReflection or reflexion may refer to: Philosophy * Self-reflection Science * Reflection (physics), a common wave phenomenon ** Specular reflection, reflection from a smooth surface *** Mirror image, a reflection in a mirror or in water ** Signal r ...
and bump-mapping and usually for
terrain rendering Terrain cartography or relief mapping is the depiction of the shape of the surface of the Earth on a map, using one or more of several techniques that have been developed. Terrain or relief is an essential aspect of physical geography, and as suc ...

. ''Outcasts
graphics engine 3.6 Rendering or image synthesis is the process of generating a physically-based rendering, photorealistic or Non-photorealistic rendering, non-photorealistic image from a 2D model, 2D or 3D model by means of a computer program. The resulting ...
was mainly a combination of a ray casting (
heightmap In computer graphics, a heightmap or heightfield is a raster graphics, raster image used mainly as Discrete Global Grid in digital elevation model#Types_of_DEM, secondary elevation modeling. Each pixel stores values, such as surface elevatio ...

) engine, used to render the landscape, and a
texture mapping Texture mapping is a method for defining high frequency detail Detail(s) or The Detail(s) may refer to: Film and television * ''Details'' (film), a 2003 Swedish film * ''The Details'' (film), a 2011 American film * '' The Detail'', a Canadian ...

polygon In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...
engine used to render objects. The "Engine Programming" section of the games credits in the manual has several subsections related to graphics, among them: "Landscape Engine", "Polygon Engine", "Water & Shadows Engine" and "Special effects Engine". Although ''Outcast'' is often cited as a forerunner of voxel technology, this is somewhat misleading. The game does not actually model three-dimensional volumes of voxels. Instead, it models the ground as a surface, which may be seen as being made up of voxels. The ground is decorated with objects that are modeled using texture-mapped polygons. When ''Outcast'' was developed, the term "voxel engine", when applied to computer games, commonly referred to a ray casting engine (for example the '' VoxelSpace'' engine). On the engine technology page of the game's website, the landscape engine is also referred to as the "Voxels engine". The engine is purely
software Software is a collection of instructions Instruction or instructions may refer to: Computing * Instruction, one operation of a processor within a computer architecture instruction set * Computer program, a collection of instructions Music * I ...

-based; it does not rely on hardware-acceleration via a 3D
graphics card A graphics card (also called a video card, display card, graphics adapter, video adapter, or display adapter) is an expansion card Modern EEPROM chip suitable for storing expansion card configuration electronically In computing Compu ...
.Voxel terrain engine
", introduction. In a coder's mind, 2005.
John Carmack John D. Carmack II (born August 20, 1970) is an American computer programmer, video game developer and engineer Engineers, as practitioners of engineering, are Professional, professionals who Invention, invent, design, analyze, build and t ...

also experimented with Voxels for the '' Quake III'' engine. One such problem cited by Carmack was the lack of graphics cards designed specifically for such rendering requiring them to be software rendered.
Comanche The Comanche or Nʉmʉnʉʉ ( com, Nʉmʉnʉʉ; "the people") are a Native American Native Americans may refer to: Ethnic groups * Indigenous peoples of the Americas, the pre-Columbian peoples of North and South America and their descendants ...
was also the first commercial
flight simulation A flight simulator is a device that artificially re-creates aircraft flight Flight or flying is the process by which an object (physics), object motion (physics), moves through a space without contacting any planetary surface, either within ...
based on voxel technology.
NovaLogic NovaLogic, Inc. was a software developer and publisher established in 1985 and based in Calabasas, California. The company was founded by CEO John A. Garcia. Garcia's background in computer software started in Southern California in the early 198 ...

used the proprietary
Voxel Space Voxel Space was a voxel raster graphics rendering engine invented by Novalogic developer and vice-president of technology, Kyle Freeman. The company was issued a patent for the technology in early 2000. History The original Voxel Space engine wa ...
engine developed for the company by Kyle Freeman (written entirely in
Assembly language In computer programming Computer programming is the process of designing and building an executable computer program to accomplish a specific computing result or to perform a particular task. Programming involves tasks such as analysis, gene ...
) to create open landscapes. This rendering technique allowed for much more detailed and realistic terrain compared to simulations based on
vector graphics Vector graphics, as a form of computer graphics, is the set of mechanisms for creating visual images directly from Geometric primitive , geometric shapes defined on a Cartesian coordinate system, Cartesian plane, such as Point (geometry), po ...
at that time.

## Gallery

3D rendering of a micro CT scan of a piece of dried leaf..ogv, 3D rendering of a µCT scan of a leaf piece, resolution circa 40 µm/voxel when viewed at the full size Ribo-Voxels.png, A (smoothed) rendering of a
data set A data set (or dataset) is a collection of data Data (; ) are individual facts, statistics, or items of information, often numeric. In a more technical sense, data are a set of values of qualitative property, qualitative or quantity, quantit ...
of voxels for a
macromolecule A macromolecule is a very large molecule A molecule is an electrically Electricity is the set of physical phenomena associated with the presence and motion Image:Leaving Yongsan Station.jpg, 300px, Motion involves a change in ...

# Data

A voxel represents a single sample, or data point, on a regularly spaced, three-dimensional grid. This data point can consist of a single piece of data, such as an opacity, or multiple pieces of data, such as a color in addition to opacity. A voxel represents only a single point on this grid, not a volume; the space between each voxel is not represented in a voxel-based dataset. Depending on the type of data and the intended use for the dataset, this missing information may be reconstructed and/or approximated, e.g. via interpolation. The value of a voxel may represent various properties. In scans, the values are
Hounsfield unitsThe Hounsfield scale , named after Sir Godfrey Hounsfield, is a quantitative scale for describing radiodensity. It is frequently used in CT scan A CT scan or computed tomography scan (formerly known as computed axial tomography or CAT scan) is a ...
, giving the opacity of material to X-rays.Novelline, Robert. ''Squire's Fundamentals of Radiology''. Harvard University Press. 5th edition. 1997. . Different types of value are acquired from
MRI Magnetic resonance imaging (MRI) is a medical imaging Medical imaging is the technique and process of imaging Imaging is the representation or reproduction of an object's form; especially a visual representation (i.e., the formation of a ...
or
ultrasound Ultrasound is sound wave In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, ...
. Voxels can contain multiple
scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers *Scalar (physics), a physical quantity that can be described by a single element of a number field such as ...
values, essentially vector (tensor) data; in the case of
ultrasound Ultrasound is sound wave In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, ...

scans with B-mode and Doppler data,
density The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its per unit . The symbol most often used for density is ''ρ'' (the lower case Greek letter ), although the Latin letter ''D'' can also ...

, and
volumetric flow rate In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ...
are captured as separate channels of data relating to the same voxel positions. While voxels provide the benefit of precision and depth of reality, they are typically large data sets and are unwieldy to manage given the bandwidth of common computers. However, through efficient compression and manipulation of large data files, interactive visualization can be enabled on consumer market computers. Other values may be useful for immediate 3D rendering, such as a surface
normal vector In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

and
color Color (American English American English (AmE, AE, AmEng, USEng, en-US), sometimes called United States English or U.S. English, is the set of varieties of the English language native to the United States. Currently, American Engli ...

. Technologies to extend voxels into 4 and 5 dimensions of data are under investigation.

# Uses

Common uses of voxels include volumetric imaging in medicine and representation of terrain in games and simulations. Voxel terrain is used instead of a
heightmap In computer graphics, a heightmap or heightfield is a raster graphics, raster image used mainly as Discrete Global Grid in digital elevation model#Types_of_DEM, secondary elevation modeling. Each pixel stores values, such as surface elevatio ...

because of its ability to represent overhangs, caves, arches, and other 3D terrain features. These concave features cannot be represented in a heightmap due to only the top 'layer' of data being represented, leaving everything below it filled (the volume that would otherwise be the inside of the caves, or the underside of arches or overhangs).

## Computer games

* '' 3D Dot Game Heroes'' uses voxels to present retro-looking graphics. * '' 7 Days to Die'' is a voxel-based open world survival horror game developed by The Fun Pimps Entertainment. * ''
Ace of Spades The Ace of Spades (also known as the Spadille and Death Card ) is traditionally the highest and most valued card in the deck of playing cards in English-speaking countries. The actual value of the card varies from game to game. Design The ornate ...
'' used
Ken Silverman Ken Silverman (born November 1, 1975) is an American game programmer, best known for writing the Build (game engine), Build engine. It was most notably utilized by ''Duke Nukem 3D'', ''Shadow Warrior'', ''Blood (video game), Blood'', and more tha ...

's Voxlap engine before being rewritten in a bespoke
OpenGL OpenGL (Open Graphics Library) is a cross-language, cross-platform In computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithm ...
engine. * ''
Amok Running amok, sometimes referred to as simply amok or having gone amok, also spelled amuck or amuk, is the act of behaving disruptively or uncontrollably. The word derives from Southeast Asian Austronesian languages (especially malay), traditionall ...
'' uses voxels for its environments. * ''
Armored Fist 3 ''Armored Fist 3'' is a Vehicle simulation game#Tank and mech simulators, tank simulation video game developed and published by NovaLogic, released on October 12, 1999. Gameplay It uses voxels, volumetric pixels, for in-game graphics, much like ...
'' is a computer game made by
Novalogic NovaLogic, Inc. was a software developer and publisher established in 1985 and based in Calabasas, California. The company was founded by CEO John A. Garcia. Garcia's background in computer software started in Southern California in the early 198 ...

that used voxel-based rendering technology. * ''
Blade Runner ''Blade Runner'' is a 1982 science fiction film Science fiction (or sci-fi) is a film genre A film genre is a Genre, stylistic or thematic category for Film, motion pictures based on similarities either in the narrative , narrative elem ...

'' is a video game that used voxels to render characters and artifacts. *'' Castle Story'', a castle building Real Time Strategy game in development, has terrain consisting of smoothed voxels * '' Comanche series'', a series of computer games made by
NovaLogic NovaLogic, Inc. was a software developer and publisher established in 1985 and based in Calabasas, California. The company was founded by CEO John A. Garcia. Garcia's background in computer software started in Southern California in the early 198 ...

that used the
Voxel Space Voxel Space was a voxel raster graphics rendering engine invented by Novalogic developer and vice-president of technology, Kyle Freeman. The company was issued a patent for the technology in early 2000. History The original Voxel Space engine wa ...
voxel rasterization for terrain rendering. * '' Command & Conquer: Tiberian Sun'', '' Command & Conquer: Red Alert 2'' and '' Command & Conquer: Yuri's Revenge'' are computer games made by
Westwood Studios Westwood Studios, Inc. was an American video game developer A video game developer is a software developer specializing in video game development – the process and related disciplines of creating video games. A game developer can range from on ...
that used voxels to render most vehicles. * ''
Crysis ''Crysis'' is a first-person shooter First-person shooter (FPS) is a video game genre, sub-genre of shooter video games centered on gun and other weapon-based combat in a First person (video games), first-person perspective, with the player ...
'', as well as ''
Cryengine 2 CryEngine (officially stylized as CRYENGINE) is a game engine designed by the German game developer Crytek Crytek GmbH is a German video game developer and software developer based in Frankfurt. Founded by the Yerli brothers in Coburg in 19 ...
'' and ''
Cryengine 3 CryEngine (officially stylized as CRYENGINE) is a game engine designed by the German game developer Crytek Crytek GmbH is a German video game developer and software developer based in Frankfurt. Founded by the Yerli brothers in Coburg in 19 ...
'', use a combination of
heightmaps In computer graphics Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialize ...
and voxels for their terrain systems. * ''
Cube World ''Cube World'' is an action role-playing game developed and published by Picroma for Microsoft Windows. Wolfram von Funck, the game's designer, began developing the game in June 2011, and was later joined by his wife, Sarah. An Software release l ...

'', an
action role-playing game ACTION is a bus operator in Canberra Canberra ( ) is the capital city of Australia. Founded following the Federation of Australia, federation of the colonies of Australia as the seat of government for the new nation, it is Australia's ...
, features procedurally generated voxel landscapes. *''
Delta Force The 1st Special Forces Operational Detachment–Delta (1st SFOD-D), commonly referred to as Delta Force, Combat Applications Group (CAG), "The Unit", or within JSOC as Task Force Green,In a 2010 article, Marc Ambinder reported that Army Compar ...
'' is a computer game made by
Novalogic NovaLogic, Inc. was a software developer and publisher established in 1985 and based in Calabasas, California. The company was founded by CEO John A. Garcia. Garcia's background in computer software started in Southern California in the early 198 ...

that used the in-house
Voxel Space Voxel Space was a voxel raster graphics rendering engine invented by Novalogic developer and vice-president of technology, Kyle Freeman. The company was issued a patent for the technology in early 2000. History The original Voxel Space engine wa ...
rendering engine. * '''' is a single-shard universe MMORPG game that uses voxels to build spaceships, civilizations, and space stations. * '' EverQuest Next'' and ''EverQuest Next: Landmark'', cancelled MMORPGs by
Sony Online Entertainment Daybreak Game Company LLC is an American video game developer A video game developer is a software developer specializing in video game development – the process and related disciplines of creating video games. A game developer can range fro ...
, make extensive use of voxels for world creation as well as player generated content *'' Hexplore'', a multi-player role playing game, uses a voxel engine allowing the player to rotate the isometric rendered playfield. * Hytale is a RPG Sandbox voxel-based game which gets developed by Hypixel Studios and gets funded by
Riot Games Riot Games, Inc. is an American video game developer A video game developer is a software developer A computer programmer, sometimes called a software developer, a programmer or more recently a coder (especially in more informal context ...
, release date is set to 2023 at earliest. * ''
Master of Orion III ''Master of Orion III'' is a 4X turn-based strategy game and the third in the ''Master of Orion'' series. ''Master of Orion III'' was developed by Quicksilver Software and published by Infogrames on February 25, 2003. Backstory In the ''Master o ...
'' uses voxel graphics to render space battles and solar systems. Battles displaying 1000 ships at a time were rendered slowly on computers without hardware graphic acceleration. * ''
Medieval Engineers ''Medieval Engineers'' is a voxel In 3D computer graphics, a voxel represents a value on a regular grid in three-dimensional space Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which ...
'' is the second voxel-based engineering game by Keen Software House inspired from medieval technology. * ''
Minecraft ''Minecraft'' is a sandbox video game A sandbox game is a video game A video game is an electronic game that involves interaction with a user interface or input device such as a joystick, game controller, controller, computer keyboard, ...

'' is a
sandbox A sandbox is a sandpit A sandpit (most Commonwealth countries) or sandbox (US and Canada) is a low, wide container or shallow depression filled with soft (beach) in which can play. Sharp sand (as used in the building industry) is n ...
video game that uses voxels to store terrain
data Data (; ) are individual facts A fact is something that is truth, true. The usual test for a statement of fact is verifiability—that is whether it can be demonstrated to correspond to experience. Standard reference works are often used ...
Novalogic NovaLogic, Inc. was a software developer and publisher established in 1985 and based in Calabasas, California. The company was founded by CEO John A. Garcia. Garcia's background in computer software started in Southern California in the early 198 ...

that used voxel-based rendering technology. It was originally bundled and shipped with the ''Comanche 2'' game although they were two separate games. * ''Worms 4: Mayhem'' uses a voxel-based engine to simulate land deformation similar to the older 2D Worms games. *Minetest uses them for Graphics pipeline#Concept, rendering nodes

# Editors

While scientific volume visualization does not require modifying the actual voxel data, voxel editors can be used to create art (especially 3D pixel art) and models for voxel based games. Some editors are focused on a single approach to voxel editing while others mix various approaches. Some common approaches are: * Slice based: The volume is sliced in one or more axes and the user can edit each image individually using Raster graphics editor, 2D raster editor tools. These generally store color information in voxels. * Sculpture: Similar to the Digital sculpting, vector counterpart but with no topology constraints. These usually store density information in voxels and lack color information. * Building blocks: The user can add and remove blocks just like a construction set toy. There are a few voxel editors available that are not tied to specific games or engines. They can be used as alternatives or complements to traditional 3D vector modeling.

# Extensions

A generalization of a voxel is the ''toxel'', or temporal voxel. This is used in the case of a Four-dimensional space, 4D dataset, for example, an image sequence that represents 3D space together with another dimension such as time. In this way, an image could contain 100×100×100×100 toxels, which could be seen as a series of 100 Film frame, frames of a 100×100×100 volume image (the equivalent for a 3D image would be showing a 2D cross section of the image in each frame). Although storage and manipulation of such data requires large amounts of memory, it allows the representation and analysis of spacetime systems.