Isotropy

<h2 id="mathematics">Mathematics</h2>
<p>Within <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, <i>isotropy</i> has a few different meanings:
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<a href="/facts/Isotropic_manifold/CUmHV45Y">Isotropic manifolds</a>
A <a href="/facts/Manifold/rXPERJtu">manifold</a> is isotropic if the <a href="/facts/Geometry/wTQf5ffQ">geometry</a> on the manifold is the same regardless of direction. A similar concept is <a href="/facts/Homogeneous_space/YGebqcJA">homogeneity</a>.
<a href="/facts/Isotropic_quadratic_form/vMMamxqE">Isotropic quadratic form</a>
A <a href="/facts/Quadratic_form/l2n1jXPe">quadratic form</a> <i>q</i> is said to be isotropic if there is a non-zero vector <i>v</i> such that <i>q</i>(<i>v</i>) = 0; such a <i>v</i> is an <a href="/facts/Isotropic_vector/sjkdQtdS">isotropic vector</a> or null vector. In complex geometry, a line through the origin in the direction of an isotropic vector is an <a href="/facts/Isotropic_line/eMeRF6mB">isotropic line</a>.
<a href="/facts/Isotropic_coordinates/VVpstuNV">Isotropic coordinates</a>
Isotropic coordinates are coordinates on an isotropic chart for <a href="/facts/Lorentzian_manifolds/VFXsVjqb">Lorentzian manifolds</a>.
<a href="/facts/Isotropy_group/Vh9VtUUw">Isotropy group</a>
An isotropy group is the group of <a href="/facts/Isomorphism/pj8KgkxU">isomorphisms</a> from any <a href="/facts/Object_(category_theory)/7WzxUThf">object</a> to itself in a <a href="/facts/Groupoid/hvj8wvWe">groupoid</a>.[<i>dubious – discuss</i>]<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> An <a href="/facts/Isotropy_representation/DMnWaql9">isotropy representation</a> is a representation of an isotropy group.
<a href="/facts/Isotropic_position/3Pm3SXXr">Isotropic position</a>
A <a href="/facts/Probability_distribution/EpsKKVRu">probability distribution</a> over a <a href="/facts/Vector_space/M1pxMLx2">vector space</a> is in isotropic position if its <a href="/facts/Covariance_matrix/kLhgjHC6">covariance matrix</a> is the <a href="/facts/Identity_matrix/Glbcf6ol">identity</a>.
Isotropic vector field
The <a href="/facts/Vector_field/ccFNuPPp">vector field</a> generated by a point source is said to be <i>isotropic</i> if, for any spherical neighborhood centered at the point source, the magnitude of the vector determined by any point on the sphere is invariant under a change in direction. For an example, starlight appears to be isotropic.
<h2 id="physics">Physics</h2>
<a href="/facts/Quantum_mechanics/BRc3Mzgr">Quantum mechanics</a> or <a href="/facts/Particle_physics/YgIuyj2t">particle physics</a>
When a spinless particle (or even an unpolarized particle with spin) decays, the resulting decay distribution <i>must</i> be isotropic in the <a href="/facts/Rest_frame/gFkIWgXk">rest frame</a> of the decaying particle - regardless of the detailed physics of the decay. This follows from <a href="/facts/Rotational_invariance/hBD8YYKx">rotational invariance</a> of the <a href="/facts/Hamiltonian_mechanics/2ReerlNA">Hamiltonian</a>, which in turn is guaranteed for a spherically symmetric potential.
Gases
The <a href="/facts/Kinetic_theory_of_gases/WUaJuMHU">kinetic theory of gases</a> also exemplifies isotropy. It is assumed that the molecules move in random directions and as a consequence, there is an equal probability of a molecule moving in any direction. Thus when there are many molecules in the gas, with high probability there will be very similar numbers moving in one direction as any other, demonstrating approximate isotropy.
<a href="/facts/Fluid_dynamics/y7uzSCfg">Fluid dynamics</a>
Fluid flow is isotropic if there is no directional preference (e.g. in fully developed 3D turbulence). An example of anisotropy is in flows with a background density as gravity works in only one direction.  The apparent surface separating two differing isotropic fluids would be referred to as an isotrope.
<a href="/facts/Thermal_expansion/BP2vjX0N">Thermal expansion</a>
A solid is said to be isotropic if the expansion of solid is equal in all directions when thermal energy is provided to the solid.
<a href="/facts/Electromagnetism/oAkmALHT">Electromagnetics</a>
An isotropic medium is one such that the <a href="/facts/Permittivity/Sd3WtgT1">permittivity</a>, ε, and <a href="/facts/Permeability_(electromagnetism)/GA7CcRLU"> permeability</a>, μ, of the medium are uniform in all directions of the medium, the simplest instance being free space.
<a href="/facts/Optics/p0vq4S1T">Optics</a>
Optical isotropy means having the same optical properties in all directions. The individual <a href="/facts/Reflectance/TT4sBwIw">reflectance</a> or <a href="/facts/Transmittance/xlyCG7ks">transmittance</a> of the domains is averaged for micro-heterogeneous samples if the macroscopic reflectance or transmittance is to be calculated. This can be verified simply by investigating, for example, a <a href="/facts/Polycrystalline/nXSgR0wp">polycrystalline</a> material under a polarizing microscope having the polarizers crossed: If the crystallites are larger than the resolution limit, they will be visible.

<a href="/facts/Cosmology/yM19zM7Y">Cosmology</a>
The <a href="/facts/Cosmological_principle/snaj0VNu">cosmological principle</a>, which underpins much of modern cosmology (including the <a href="/facts/Big_Bang/I5T1vTnv">Big Bang</a> theory of the evolution of the observable universe), assumes that the universe is both isotropic and homogeneous, meaning that the universe has no preferred location (is the same everywhere) and has no preferred direction.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>  Observations[<i>which?</i>] made in 2006 suggest that, on distance-scales much larger than <a href="/facts/Galaxy/zCyUvEf7"> galaxies</a>, <a href="/facts/Galaxy_cluster/BhIEgbx8">galaxy clusters</a> are <a href="/facts/Great_Wall_(astronomy)/r8XpATsh">"Great"</a> features, but small compared to so-called <a href="/facts/Multiverse/GUCDxvPg">multiverse</a> scenarios.
<h3>Materials science</h3>

<p>In the study of <a href="/facts/List_of_materials_properties/Cg2LVoQe">mechanical properties of materials</a>, "isotropic" means having identical values of a property in all directions.  This definition is also used in <a href="/facts/Geology/WGvVRIYF">geology</a> and <a href="/facts/Mineralogy/MeIK9lx9">mineralogy</a>. Glass and metals are examples of isotropic materials.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>  Common anisotropic materials include <a href="/facts/Wood/z13NpHPl">wood</a> (because its material properties are different parallel to and perpendicular to the grain) and layered rocks such as <a href="/facts/Slate/hSVHL6Lj">slate</a>.
</p><p>Isotropic materials are useful since they are easier to shape, and their behavior is easier to predict. Anisotropic materials can be tailored to the forces an object is expected to experience. For example, the fibers in <a href="/facts/Carbon_fiber/6PlL4mB2">carbon fiber</a> materials and <a href="/facts/Rebar/qm27fUbk">rebars</a> in <a href="/facts/Reinforced_concrete/aUTUKAQF">reinforced concrete</a> are oriented to withstand tension.
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<h3><a href="/facts/Etching_(microfabrication)/kPa01dgL"> Microfabrication</a></h3>
<p>In industrial processes, such as <a href="/facts/Etching/PM0FmoSb">etching</a> steps, "isotropic" means that the process proceeds at the same rate, regardless of direction. Simple chemical reaction and removal of a substrate by an acid, a solvent or a reactive gas is often very close to isotropic. Conversely, "anisotropic" means that the attack rate of the substrate is higher in a certain direction. Anisotropic etch processes, where vertical etch-rate is high but lateral etch-rate is very small, are essential processes in <a href="/facts/Microfabrication/TzGo63pr">microfabrication</a> of <a href="/facts/Integrated_circuits/fjn6vg7C">integrated circuits</a> and <a href="/facts/Microelectromechanical_systems/TkhEzDpm">MEMS</a> devices.
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<h3>Antenna (radio)</h3>
<p>An <a href="/facts/Isotropic_antenna/HFGgrM74">isotropic antenna</a> is an idealized <a href="/facts/Radiator/W592jFf2"> "radiating element"</a> used as a <a href="/facts/Reference/pLmVxBjJ">reference</a>; an <a href="/facts/Antenna_(electronics)/pGlcXBvc">antenna</a> that broadcasts power equally (calculated by the <a href="/facts/Poynting_vector/Er8YIIDk">Poynting vector</a>) in all directions. The <a href="/facts/Antenna_gain/ke1y0LcM">gain</a> of an arbitrary antenna is usually reported in <a href="/facts/Decibel/0LyxAPXh">decibels</a> relative to an isotropic antenna, and is expressed as <a href="/facts/DBi/0LyxAPXh">dBi</a> or dB(i).
</p><p>In cells (a.k.a. <a href="/facts/Muscle_fibers/cP7Jea3q">muscle fibers</a>), the term <a href="/facts/Isotropic_bands/mcn0DVoF"> "isotropic"</a> refers to the light bands (<a href="/facts/I_bands/9jQ5DaTz">I bands</a>) that contribute to the striated pattern of the cells.
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<h3><a href="/facts/Pharmacology/4Rx8s7Ie">Pharmacology</a></h3>
<p>While it is well established that the skin provides an ideal site for the administration of local and systemic drugs, it presents a formidable barrier to the permeation of most substances.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> Recently, <a href="/facts/Isotropic_formulations/ImpX967a">isotropic formulations</a> have been used extensively in dermatology for drug delivery.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a>
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<h2 id="computer-science">Computer science</h2>
<a href="/facts/Medical_imaging/pL6JyJEp">Imaging</a>
A volume such as a <a href="/facts/Computed_tomography/NusPJ2S8">computed tomography</a> is said to have isotropic <a href="/facts/Voxel/xo077lPE">voxel</a> spacing when the space between any two adjacent voxels is the same along each axis <i>x, y, z</i>. E.g., voxel spacing is isotropic if the center of voxel <i>(i, j, k)</i> is 1.38 mm from that of <i>(i+1, j, k)</i>, 1.38 mm from that of <i>(i, j+1, k)</i> and 1.38 mm from that of <i>(i, j, k+1)</i> for all indices <i>i, j, k</i>.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a>
<h2 id="other-sciences">Other sciences</h2>
<a href="/facts/Economics/HYctA2Dn">Economics</a> and <a href="/facts/Geography/TRd1Toyo">geography</a>
An isotropic region is a region that has the same properties everywhere. Such a region is a construction needed in many types of models.
<h2 id="see-also">See also</h2>

Look up <i>isotropy</i> in Wiktionary, the free dictionary.

<ul><li><a href="/facts/Rotational_invariance/hBD8YYKx">Rotational invariance</a></li>
<li><a href="/facts/Isotropic_bands/mcn0DVoF">Isotropic bands</a></li>
<li><a href="/facts/Isotropic_coordinates/VVpstuNV">Isotropic coordinates</a></li>
<li><a href="/facts/Transverse_isotropy/tMKjMH7J">Transverse isotropy</a></li>
<li><a href="/facts/Bi_isotropic/aJ2MsL5p">Bi isotropic</a></li>
<li><a href="/facts/Symmetry/hpu7DK8p">Symmetry</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>A groupoid 
  
    
      
        
          
            G
          
        
      
    
    {\displaystyle {\mathcal {G}}}
  
 is a category where all morphisms are isomorphisms, i.e., invertible. If 
  
    
      
        G
        ∈
        
          
            G
          
        
      
    
    {\displaystyle G\in {\mathcal {G}}}
  
 is any object, then 
  
    
      
        
          
            G
          
        
        (
        G
        ,
        G
        )
      
    
    {\displaystyle {\mathcal {G}}(G,G)}
  
 denotes its isotropy group: the group of isomorphisms from 
  
    
      
        G
      
    
    {\displaystyle G}
  
 to 
  
    
      
        G
      
    
    {\displaystyle G}
  
. <a href="/wiki/Groupoid" target="_blank">/wiki/Groupoid</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>"WMAP Big Bang Theory". Map.gsfc.nasa.gov. Retrieved 2014-03-06. <a href="https://map.gsfc.nasa.gov/universe/bb_theory.html" target="_blank">https://map.gsfc.nasa.gov/universe/bb_theory.html</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>"Anisotropy and Isotropy". Archived from the original on 2012-05-31. Retrieved 2012-05-26. <a href="https://web.archive.org/web/20120531172526/http://www.ndt-ed.org/EducationResources/CommunityCollege/Materials/Structure/anisotropy.htm" target="_blank">https://web.archive.org/web/20120531172526/http://www.ndt-ed.org/EducationResources/CommunityCollege/Materials/Structure/anisotropy.htm</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Landman L. "The Epidermal Permeability Barrier". Anatomy and Embryology (Berl) 1988; 178:1-13 [1] <a href="/wiki/Anatomy_and_Embryology" target="_blank">/wiki/Anatomy_and_Embryology</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Gregoriadis G. "Liposomes in Drug Delivery".  Harwood Academic Publishers, 1993. [2] <a href="https://books.google.com/books?id=IqqYFne6deEC&dq=Liposomes+in+Drug+Delivery&pg=PR7" target="_blank">https://books.google.com/books?id=IqqYFne6deEC&dq=Liposomes+in+Drug+Delivery&pg=PR7</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Zwanenburg, Alex; Leger, Stefan; Vallières, Martin; Löck, Steffen (2016-12-21). "Image biomarker standardisation initiative". Radiology. 295 (2): 328–338. arXiv:1612.07003. doi:10.1148/radiol.2020191145. PMC 7193906. PMID 32154773. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7193906" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7193906</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
</ol>

Isotropy open-in-new

Isotropy