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I've been looking for an explanation of why the circles of the Hopf fibration become linked. This is a request for someone more knowledgeable to fill in this missing information - Gauge 17:51, 2 Apr 2005 (UTC)
This is a somewhat flaky question, but ... I'm wondering if there's a "natural" metric associated with a Hopf fibration. The "natural" metric on CP^n is the Fubini-Study metric, which is identical to the ordinary metric on the two-sphere for CP^1. I can certainly pullback the metric on S^2 to define a metric on S^3, but I'm wondering how "natural" this really is, if it has any interesting non-intuitive or enligtening properties.
For example, if I envision S^3 as the EUcliden space R^3 that we live in, with an extra point at infinity, then the Hopf fibration fills this space with non-intersection circles (as illstrated by the "keyring fibration" photo). Each circle has a center ... what is the density of the distribution of the centers of these circles in R^3, (assuming a uniform density on S^2)? Are the centers of these circles always confined to a plane? What is the distribution on the plane? Uniform? Gaussian? Each circle defines a direction (the normal to the plane containing the circle). What is the distribution of these directions? linas 16:23, 26 June 2006 (UTC)
While I sympathize with the aim of beginning articles with accessible language, the claim that "the Hopf bundle (or Hopf fibration) … is a partition of a 3-dimensional hypersphere into circles" misrepresents the essential mathematics.
Yes, a fiber bundle has fibers, but the topological relationship between the base space and the total space through the projection map is what makes it important. In particular, if we look at the inverse image of a neighborhood in the base, that portion of the bundle looks like a product of the neighborhood and the fiber space. This "local product space" structure is what allows us to do, say, path lifting.
Better pictures of the Hopf bundle suggest this topology by showing nested tori, not just circles. Some of the earliest computer graphics instances are the work of Thomas Banchoff, whose "flat torus" is the inverse image of a circle of S2. And he shows circle geometry, not just topology, because the image uses stereographic projection from S3 to R3. A visualization of the entire bundle, not just one torus, can be found at the Hopf Topology Archive. Follow the link from the main page to see an image using colors on both S3 and S2, and other strategems, to reveal structure. (It is also found in the SIGGRAPH 94 Art and Design Slide Set, and in Graphics Gems IV.) In this one the circles are only topological, but are confined to a finite ball.
The (pre-existing) keyrings "model" in the picture leading the article is as unhelpful as the "partition" prose, though it has other appeal. (The better images I mentioned cannot be used because of copyright.) Also, I'm afraid the "One topological model" sentence is a move in the wrong direction, especially for the lay reader, for whom it will be gibberish.
So, care to try again? -- KSmrq T 22:20, 8 October 2007 (UTC)
I'm happy to have my prose dissected and improved. I'd like to defend the "one topological model" part, though: the point was to describe the bundle in terms lay readers might hope to be able to visualize (circles in the one-point completion of 3d) rather than leaving them with the impression that as an object involving abstract 3-manifolds it is unvisualizable. Similarly, while the local product structure is essential to the mathematical content, I don't think it's essential to a lead that gives lay readers some idea of what this is about. — David Eppstein 23:34, 8 October 2007 (UTC)
{{
citation}}
form so I could get the automatic links from {{
harv}}
, and expanded them a little.Why is this article at "Hopf bundle" instead of "Hopf fibration" anyway? -- Horoball 17:39, 9 October 2007 (UTC)
I found some notes I made while studying how to fill space with nested toruses made of Villarceau circles. For the benefit of others who may wish to experiment:
Since φ merely rotates the circles around the z axis, the general center and plane are easily obtained. In the limiting cases for θ, the torus degenerates to the unit circle in the xy plane when θ = 0 and to the z axis when θ = π. -- KSmrq T 10:24, 11 October 2007 (UTC)
The sentence "discovered by Hopf in 1931" is what's in question.
In "Such silver currents,..." the biography of W.K. Clifford, it says Clifford discovered the "Hopf" fibration, and that Hopf was more scrupulous than anyone in giving credit to Clifford. Rather than trying to dig up 100-year-old references, does anyone here know more details of this? Perhaps KSmrq, who might be referring not just to conceptual importance of the locally trivial aspect, but also to some historical importance as well. Is the fibration nature of this example due to Hopf? —Preceding unsigned comment added by 137.146.194.173 ( talk) 19:27, 12 June 2009 (UTC)
Example from the fluid mechanics looks strange: from equations
it follows that
id est,
To me, such a relation looks strange, is this valid for some realistic fluid? Shouldn't $A$ and $B$ be real constant? Shouldn't $p$ and $\rho$ be positive? Shouldn't pressure $p$ increase with increase of density $\rho$? dima ( talk) 04:07, 2 September 2012 (UTC)
The recent addition of the 600-cell, "The 600-cell partitions into 20 rings of 30 tetrahedra each." needs some clarification. The tetrahedron does not have opposing parallel faces, so there is no way you can stack these "end-to-end" in a great circle. It necessarily has geodesic curvature. At best you can make a closed chain that has some helicity, whos axis would be a great circle. Still very interesting. Cloudswrest ( talk) 23:24, 2 October 2012 (UTC)
“ | The 600-cell also partitions into 20 rings of 30 tetrahedra each in a very interesting, quasi-regular chain called the Boerdijk–Coxeter helix. When superimposed onto the 3-sphere curvature it becomes regular. | ” |
What do "regular" and "quasi-regular" mean in this context? — Tamfang ( talk) 19:18, 12 October 2012 (UTC)
I think the article is excellent but that there is still room for improvement.
Two of the illustrations are, I think, misleading:
a) One is the beautiful multicolor picture of the Hopf fibration S3 → S2 at the upper right, of a snail-shellish surface that's a union of Hopf fibres. It is certainly true that one can find such a snail-shellish surface that is the union of Hopf fibres. But a much simpler way of describing this Hopf fibration is used in the text: a union of tori, all having the same core circle, and each filled (foliated) with a family of circles of Villarceau (which are all congruent to the common core circle.
Also, the caption of this illustration reads in part: "The Hopf fibration can be visualized using a stereographic projection of S3 to R3 and then compressing R3 to a ball." I don't know how this can be done as stated other than in a very distorted manner.
Rather, after *removing* one of the fibres in S3 one can arrange that the image be R3 minus the z-axis, and *this* can be compressed to a maximal open solid torus of revolution that shows clearly the "concentric" tori and their circles of Villarceau.
b) The illustration of the circles of Villarceau is certainly accurate. But it is also extremely misleading in the context of the Hopf fibration S3 → S2, since it shows the two circles intersecting! Of course, this never happens in a fibre bundle. This illustration is perfectly appropriate for the article on circles of Villarceau. But here it is not. It would be much better if a picture showed clearly how a family of (disjoint) circles of Villarceau can fill up a torus of revolution.
c) In the text, neither the section titled "Geometry and applications" nor any other section seems to mention anything about the radius of the base space in a Hopf fibration. If the Hopf fibration has as total space a sphere S2k-1 (k = 1,2,3, or 4) of radius = 1, with the usual action of the unit (reals, complexes, quaternions, or octonions) or in other words, the usual way to define P1, CP1, HP1, or OP1, respectively, as the quotient of this action.
The radius of the base spheres in the Hopf fibrations can be seen to be = 1/2. This ought to be stated. Daqu ( talk) 21:31, 29 August 2013 (UTC)
It says "... a 3-sphere (a hypersphere in four-dimensional space)..". Is that non-generic description within brackets really necessary? - Subh83 ( talk | contribs) 07:01, 2 January 2014 (UTC)
Where can I find the image of a stereographic projection of S0-S1-S1? 67.243.159.27 ( talk) 12:31, 24 February 2014 (UTC)
Should there be a mention of Bowers' polytwisters (and polytwirlers/polywhirlers)? — Preceding unsigned comment added by 98.207.169.109 ( talk) Sep 1, 2017
I started working on a project (Github), which represents rotations with Euler Axis, Axis-Angle, or angle-angle-angle sort of representation. All rotation methods such as matrices and quaternions have a common base that is angle-axis.
These early demos... https://d3x0r.github.io/STFRPhysics/3d/index4.html (hopf fibration generation, generating a single fiber with analog steps instead of a discrete number of separate rings) https://d3x0r.github.io/STFRPhysics/3d/index.html This really shows just a single loop, which is the rotation coordinates of a base rotation, rotated around the 'yaw' of that point.
The hopf fibration generator though is simply a few loops that takes 3 rotation axis and iterates the angles that are applied for each. There's a outer loop of 0-2pi, and inner loops from -2pi to 2pi for a number of turns around... The settings are - a number of turns - which breaks up the second loop into an integer fraction... and then applied with all the appropriate fractional steps.
I had initially done it in discrete loops, so there was a for count of turns, for 0-4pi around each turn, rotate Q0 by Q1 and rotate that result around the axis Q3. I don't find any discontinuities.... so I'm skeptical of MUCH content of this whole article. Generally the resulting coordinates generally form a toroid; mind you though it's plotted in 3D, there's nothing linear about this, it's all about rotations, and the evolution from one point in time to another by compositing three rotations.
This is basically the loop code... (A,B,C)*T = R0, R1 and R2 are other unit vector axii of rotations which are used. 'freeSpin' is application of the composite rotation Rodrigues' Rotation Formula, which rotates an axis and angle around another axis and angle.
const lnQ0 = new lnQuat( 0, T*A/lATC, T*B/lATC, T*C/lATC ).update(); for( let nTotal = 0; nTotal < steps; nTotal++ ) { fibre = nTotal * ( 4*Math.PI ) / ( steps ); const fiberPart =((fibre + 1*Math.PI)/(Math.PI*2) %(1/subSteps)); const t = (Math.PI*4)* subSteps*(fiberPart) - (Math.PI*2); const lnQ = new lnQuat( lnQ0 ) // copy lnQ0 before 'spinning' .freeSpin( fibre, {x:R1.x,y:R1.y,z:R1.z} ) .freeSpin( t, {x:R2.x, y:R2.y, z:R2.z} ); // plot each point connected to each prior point // each point is scaled angle*(axis) where axis is a unit vector.
Edit: Initially I was doing it more like fibers to make individual rings, so rather than rotating the rotation, I was just iterating the axis, and then rotating that in the end; when doing it as a single path like above it's not actually correct to step the axis like that; however, for more of a Hopf Fibration like the youtube video with interlinked rings (non chaotic). but then the result betrays that half of a cycle is interlaced with the other half.
But I don't really know anything about abstract topologies; and maybe what they think is discontinuous is actually apparently continuous to the layman?
21:06, 2 April 2021 (UTC) (I thought I signed that) D3x0r ( talk) 21:07, 2 April 2021 (UTC)
---
Really I'd love to know more about this whole thing; I obviously don't have perfect understandings; but this is certainly the 'natural metric' that someone above was seeking so very long ago; but then the answer is something about a map and a space S3, which is never really 'shown' just the idea of the space... since all these nested toroidal structures sort of look like a 'metric-able' thing?
I did play with converting the normal to a 2 angle definition, and projecting that, but it's hard to recover the original 'rotation space' covered so it all collapses to a small rectangular patch (at least for the latitude-longitude sphere I was mapping) which then makes the initial state just a flat XY grid; only that only works in the original case, and that grid warps, and gets a sin curve in it when adjusted with additional 'spin' around the normal... the normals were built such that for some point on the sphere 'up' or perpendicular to the surface of the sphere is a 'normal' and 'forward' is aligned with latitude lines and 'right' is aligned along longitude lines; from that state, the same normal, has a rotation basis that is the 'spin' around that normal... but the normal itself can be represented with 2 coordinates... however, mapping back to those angles to adjust doesn't seem practical yet - the Rodrigeus Composite rotation formula is quite complex, and substituting the sin/cos of the angles to define the axis in place of the normal X/Y/Z rotation axis didn't help the situation.
D3x0r ( talk) 21:44, 2 April 2021 (UTC)
The section here needs some serious work. It claims "The formula given for p above defines an explicit diffeomorphism between the complex projective line and the ordinary 2-sphere in 3-dimensional space." - except that p is defined as a map from S^3 to S^2. I know that p descends along the quotient map S^3 -> CP^1, as it is invariant under multiplication on the domain by unit complex numbers, so in principle one could instead specify a map on CP^1 using homogeneous coordinates coming from S^3, and then claim that that is a diffeomorphism. Albeit it's not obvious this map is either injective or surjective, nor what its inverse is. 121.45.89.81 ( talk) 23:33, 19 September 2021 (UTC)
![]() | This article is rated B-class on Wikipedia's
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I've been looking for an explanation of why the circles of the Hopf fibration become linked. This is a request for someone more knowledgeable to fill in this missing information - Gauge 17:51, 2 Apr 2005 (UTC)
This is a somewhat flaky question, but ... I'm wondering if there's a "natural" metric associated with a Hopf fibration. The "natural" metric on CP^n is the Fubini-Study metric, which is identical to the ordinary metric on the two-sphere for CP^1. I can certainly pullback the metric on S^2 to define a metric on S^3, but I'm wondering how "natural" this really is, if it has any interesting non-intuitive or enligtening properties.
For example, if I envision S^3 as the EUcliden space R^3 that we live in, with an extra point at infinity, then the Hopf fibration fills this space with non-intersection circles (as illstrated by the "keyring fibration" photo). Each circle has a center ... what is the density of the distribution of the centers of these circles in R^3, (assuming a uniform density on S^2)? Are the centers of these circles always confined to a plane? What is the distribution on the plane? Uniform? Gaussian? Each circle defines a direction (the normal to the plane containing the circle). What is the distribution of these directions? linas 16:23, 26 June 2006 (UTC)
While I sympathize with the aim of beginning articles with accessible language, the claim that "the Hopf bundle (or Hopf fibration) … is a partition of a 3-dimensional hypersphere into circles" misrepresents the essential mathematics.
Yes, a fiber bundle has fibers, but the topological relationship between the base space and the total space through the projection map is what makes it important. In particular, if we look at the inverse image of a neighborhood in the base, that portion of the bundle looks like a product of the neighborhood and the fiber space. This "local product space" structure is what allows us to do, say, path lifting.
Better pictures of the Hopf bundle suggest this topology by showing nested tori, not just circles. Some of the earliest computer graphics instances are the work of Thomas Banchoff, whose "flat torus" is the inverse image of a circle of S2. And he shows circle geometry, not just topology, because the image uses stereographic projection from S3 to R3. A visualization of the entire bundle, not just one torus, can be found at the Hopf Topology Archive. Follow the link from the main page to see an image using colors on both S3 and S2, and other strategems, to reveal structure. (It is also found in the SIGGRAPH 94 Art and Design Slide Set, and in Graphics Gems IV.) In this one the circles are only topological, but are confined to a finite ball.
The (pre-existing) keyrings "model" in the picture leading the article is as unhelpful as the "partition" prose, though it has other appeal. (The better images I mentioned cannot be used because of copyright.) Also, I'm afraid the "One topological model" sentence is a move in the wrong direction, especially for the lay reader, for whom it will be gibberish.
So, care to try again? -- KSmrq T 22:20, 8 October 2007 (UTC)
I'm happy to have my prose dissected and improved. I'd like to defend the "one topological model" part, though: the point was to describe the bundle in terms lay readers might hope to be able to visualize (circles in the one-point completion of 3d) rather than leaving them with the impression that as an object involving abstract 3-manifolds it is unvisualizable. Similarly, while the local product structure is essential to the mathematical content, I don't think it's essential to a lead that gives lay readers some idea of what this is about. — David Eppstein 23:34, 8 October 2007 (UTC)
{{
citation}}
form so I could get the automatic links from {{
harv}}
, and expanded them a little.Why is this article at "Hopf bundle" instead of "Hopf fibration" anyway? -- Horoball 17:39, 9 October 2007 (UTC)
I found some notes I made while studying how to fill space with nested toruses made of Villarceau circles. For the benefit of others who may wish to experiment:
Since φ merely rotates the circles around the z axis, the general center and plane are easily obtained. In the limiting cases for θ, the torus degenerates to the unit circle in the xy plane when θ = 0 and to the z axis when θ = π. -- KSmrq T 10:24, 11 October 2007 (UTC)
The sentence "discovered by Hopf in 1931" is what's in question.
In "Such silver currents,..." the biography of W.K. Clifford, it says Clifford discovered the "Hopf" fibration, and that Hopf was more scrupulous than anyone in giving credit to Clifford. Rather than trying to dig up 100-year-old references, does anyone here know more details of this? Perhaps KSmrq, who might be referring not just to conceptual importance of the locally trivial aspect, but also to some historical importance as well. Is the fibration nature of this example due to Hopf? —Preceding unsigned comment added by 137.146.194.173 ( talk) 19:27, 12 June 2009 (UTC)
Example from the fluid mechanics looks strange: from equations
it follows that
id est,
To me, such a relation looks strange, is this valid for some realistic fluid? Shouldn't $A$ and $B$ be real constant? Shouldn't $p$ and $\rho$ be positive? Shouldn't pressure $p$ increase with increase of density $\rho$? dima ( talk) 04:07, 2 September 2012 (UTC)
The recent addition of the 600-cell, "The 600-cell partitions into 20 rings of 30 tetrahedra each." needs some clarification. The tetrahedron does not have opposing parallel faces, so there is no way you can stack these "end-to-end" in a great circle. It necessarily has geodesic curvature. At best you can make a closed chain that has some helicity, whos axis would be a great circle. Still very interesting. Cloudswrest ( talk) 23:24, 2 October 2012 (UTC)
“ | The 600-cell also partitions into 20 rings of 30 tetrahedra each in a very interesting, quasi-regular chain called the Boerdijk–Coxeter helix. When superimposed onto the 3-sphere curvature it becomes regular. | ” |
What do "regular" and "quasi-regular" mean in this context? — Tamfang ( talk) 19:18, 12 October 2012 (UTC)
I think the article is excellent but that there is still room for improvement.
Two of the illustrations are, I think, misleading:
a) One is the beautiful multicolor picture of the Hopf fibration S3 → S2 at the upper right, of a snail-shellish surface that's a union of Hopf fibres. It is certainly true that one can find such a snail-shellish surface that is the union of Hopf fibres. But a much simpler way of describing this Hopf fibration is used in the text: a union of tori, all having the same core circle, and each filled (foliated) with a family of circles of Villarceau (which are all congruent to the common core circle.
Also, the caption of this illustration reads in part: "The Hopf fibration can be visualized using a stereographic projection of S3 to R3 and then compressing R3 to a ball." I don't know how this can be done as stated other than in a very distorted manner.
Rather, after *removing* one of the fibres in S3 one can arrange that the image be R3 minus the z-axis, and *this* can be compressed to a maximal open solid torus of revolution that shows clearly the "concentric" tori and their circles of Villarceau.
b) The illustration of the circles of Villarceau is certainly accurate. But it is also extremely misleading in the context of the Hopf fibration S3 → S2, since it shows the two circles intersecting! Of course, this never happens in a fibre bundle. This illustration is perfectly appropriate for the article on circles of Villarceau. But here it is not. It would be much better if a picture showed clearly how a family of (disjoint) circles of Villarceau can fill up a torus of revolution.
c) In the text, neither the section titled "Geometry and applications" nor any other section seems to mention anything about the radius of the base space in a Hopf fibration. If the Hopf fibration has as total space a sphere S2k-1 (k = 1,2,3, or 4) of radius = 1, with the usual action of the unit (reals, complexes, quaternions, or octonions) or in other words, the usual way to define P1, CP1, HP1, or OP1, respectively, as the quotient of this action.
The radius of the base spheres in the Hopf fibrations can be seen to be = 1/2. This ought to be stated. Daqu ( talk) 21:31, 29 August 2013 (UTC)
It says "... a 3-sphere (a hypersphere in four-dimensional space)..". Is that non-generic description within brackets really necessary? - Subh83 ( talk | contribs) 07:01, 2 January 2014 (UTC)
Where can I find the image of a stereographic projection of S0-S1-S1? 67.243.159.27 ( talk) 12:31, 24 February 2014 (UTC)
Should there be a mention of Bowers' polytwisters (and polytwirlers/polywhirlers)? — Preceding unsigned comment added by 98.207.169.109 ( talk) Sep 1, 2017
I started working on a project (Github), which represents rotations with Euler Axis, Axis-Angle, or angle-angle-angle sort of representation. All rotation methods such as matrices and quaternions have a common base that is angle-axis.
These early demos... https://d3x0r.github.io/STFRPhysics/3d/index4.html (hopf fibration generation, generating a single fiber with analog steps instead of a discrete number of separate rings) https://d3x0r.github.io/STFRPhysics/3d/index.html This really shows just a single loop, which is the rotation coordinates of a base rotation, rotated around the 'yaw' of that point.
The hopf fibration generator though is simply a few loops that takes 3 rotation axis and iterates the angles that are applied for each. There's a outer loop of 0-2pi, and inner loops from -2pi to 2pi for a number of turns around... The settings are - a number of turns - which breaks up the second loop into an integer fraction... and then applied with all the appropriate fractional steps.
I had initially done it in discrete loops, so there was a for count of turns, for 0-4pi around each turn, rotate Q0 by Q1 and rotate that result around the axis Q3. I don't find any discontinuities.... so I'm skeptical of MUCH content of this whole article. Generally the resulting coordinates generally form a toroid; mind you though it's plotted in 3D, there's nothing linear about this, it's all about rotations, and the evolution from one point in time to another by compositing three rotations.
This is basically the loop code... (A,B,C)*T = R0, R1 and R2 are other unit vector axii of rotations which are used. 'freeSpin' is application of the composite rotation Rodrigues' Rotation Formula, which rotates an axis and angle around another axis and angle.
const lnQ0 = new lnQuat( 0, T*A/lATC, T*B/lATC, T*C/lATC ).update(); for( let nTotal = 0; nTotal < steps; nTotal++ ) { fibre = nTotal * ( 4*Math.PI ) / ( steps ); const fiberPart =((fibre + 1*Math.PI)/(Math.PI*2) %(1/subSteps)); const t = (Math.PI*4)* subSteps*(fiberPart) - (Math.PI*2); const lnQ = new lnQuat( lnQ0 ) // copy lnQ0 before 'spinning' .freeSpin( fibre, {x:R1.x,y:R1.y,z:R1.z} ) .freeSpin( t, {x:R2.x, y:R2.y, z:R2.z} ); // plot each point connected to each prior point // each point is scaled angle*(axis) where axis is a unit vector.
Edit: Initially I was doing it more like fibers to make individual rings, so rather than rotating the rotation, I was just iterating the axis, and then rotating that in the end; when doing it as a single path like above it's not actually correct to step the axis like that; however, for more of a Hopf Fibration like the youtube video with interlinked rings (non chaotic). but then the result betrays that half of a cycle is interlaced with the other half.
But I don't really know anything about abstract topologies; and maybe what they think is discontinuous is actually apparently continuous to the layman?
21:06, 2 April 2021 (UTC) (I thought I signed that) D3x0r ( talk) 21:07, 2 April 2021 (UTC)
---
Really I'd love to know more about this whole thing; I obviously don't have perfect understandings; but this is certainly the 'natural metric' that someone above was seeking so very long ago; but then the answer is something about a map and a space S3, which is never really 'shown' just the idea of the space... since all these nested toroidal structures sort of look like a 'metric-able' thing?
I did play with converting the normal to a 2 angle definition, and projecting that, but it's hard to recover the original 'rotation space' covered so it all collapses to a small rectangular patch (at least for the latitude-longitude sphere I was mapping) which then makes the initial state just a flat XY grid; only that only works in the original case, and that grid warps, and gets a sin curve in it when adjusted with additional 'spin' around the normal... the normals were built such that for some point on the sphere 'up' or perpendicular to the surface of the sphere is a 'normal' and 'forward' is aligned with latitude lines and 'right' is aligned along longitude lines; from that state, the same normal, has a rotation basis that is the 'spin' around that normal... but the normal itself can be represented with 2 coordinates... however, mapping back to those angles to adjust doesn't seem practical yet - the Rodrigeus Composite rotation formula is quite complex, and substituting the sin/cos of the angles to define the axis in place of the normal X/Y/Z rotation axis didn't help the situation.
D3x0r ( talk) 21:44, 2 April 2021 (UTC)
The section here needs some serious work. It claims "The formula given for p above defines an explicit diffeomorphism between the complex projective line and the ordinary 2-sphere in 3-dimensional space." - except that p is defined as a map from S^3 to S^2. I know that p descends along the quotient map S^3 -> CP^1, as it is invariant under multiplication on the domain by unit complex numbers, so in principle one could instead specify a map on CP^1 using homogeneous coordinates coming from S^3, and then claim that that is a diffeomorphism. Albeit it's not obvious this map is either injective or surjective, nor what its inverse is. 121.45.89.81 ( talk) 23:33, 19 September 2021 (UTC)