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# New in 13: Visualization & Graphics

Two years ago we released Version 12.0 of the Wolfram Language. Here are the updates in visualization and graphics since then, including the latest features in 13.0. The contents of this post are compiled from Stephen Wolfram’s Release Announcements for 12.1, 12.2, 12.3 and 13.0.

## Vector & Complex Visualization

### Cracking the Vector-Plotting Problem (March 2020)

I’ve been trying to make good vector plots for about 40 years, but in the past it just never worked. If the vectors got too short, you couldn’t see their direction—and if you made them longer they crashed into each other. But particularly after our success in Version 12.0 in cracking the ComplexPlot problem (which had also been languishing for a long time) we decided for Version 12.1 to try to solve the vector-plotting problem once and for all. And, I’m happy to say, we seem to have been able to do that.

So now, you can just ask VectorPlot (and all sorts of related functions) to make a vector plot, and you’ll automatically get something that’s a good representation of your vector field:

 ✕ `VectorPlot[{2 x^2 - y^2, 3 x y}, {x, -5, 5}, {y, -5, 5}]`
 ✕ ```VectorPlot[{2 x^2 - y^2, 3 x y}, {x, -5, 5}, {y, -5, 5}, VectorPoints -> 30]```

What’s the trick? It’s basically about placing vectors on a hexagonal grid so they’re packed better, and are visually more uniform. (You can also make other choices if you want to.) And then it’s about using appropriately scaled color to represent vector magnitudes.

There are all sorts of other challenges too. Like being able to draw vectors in a region:

 ✕ `VectorPlot[{2 x^2 - y^2, 3 x y}, {x, y} \[Element] Disk[{0, 0}, 3]]`

And putting together our complex-number-plotting capabilities with our new vector plotting, we also in 12.1 have ComplexVectorPlot:

 ✕ `ComplexVectorPlot[z^ Log[z], {z, 6}, PlotLegends -> Automatic]`

## Multipanel & Multiaxis Visualization

### Multiaxis and Multipanel Plots (December 2021)

It’s been requested for ages. And there’ve been many package implementations of it. But now in Version 13.0 we have multiaxis plotting directly built into Wolfram Language. Here’s an example:

 ✕

As indicated, the scale for the blue curve is on the left, and for the orange one on the right.

You might think this looks straightforward. But it’s not. In effect there are multiple coordinate systems all combined into one plot—and then disambiguated by axes linked by various forms of styling. The final step in the groundwork for this was laid in Version 12.3, when we introduced AxisObject and “disembodied axes”.

Here’s a more complicated case, now with 5 curves, each with its own axis:

 ✕

And here’s what happens if some curves share their axes:

 ✕

Multiple axes let you pack multiple curves onto a single “plot panel”. Multipanel plots let you put curves into separate, connected panels, with shared axes. The first cases of multipanel plots were already introduced in Version 12.0. But now in Version 13.0 we’re expanding multipanel plots to other types of visualizations:

 ✕

## Graphics Lighting, Fillers & Shaders

### Cross-Hatching & All That (March 2020)

Before there were gray scales, there were things like cross-hatching. Look at a book from a century ago (or less), and you’ll see all sorts of diagrams elegantly drawn with things like cross-hatching. Well, now we can do that too.

 ✕ `Graphics[Style[RegularPolygon[5], HatchFilling[]]]`
 ✕ ```Plot[{Sin[x], Cos[x]}, {x, 0, 10}, Filling -> Axis, FillingStyle -> HatchFilling[]]```

Of course, everything is computable:

 ✕ ```Graphics[Table[ Style[Disk[RandomReal[10, 2]], HatchFilling[RandomReal[{0, 2 \[Pi]}]]], 50]]```

We also have an important generalization of cross-hatching: PatternFilling. Here are examples with named patterns:

 ✕ `Graphics[Style[Disk[], PatternFilling["DiamondBox"]]]`
 ✕ `Graphics[Style[Disk[], PatternFilling[{"Checkerboard", Red, Black}]]]`

You can use any image as a pattern too:

 ✕ ```GeoGraphics[ Style[Polygon[Entity["Country", "UnitedStates"]], PatternFilling[CloudGet["https://wolfr.am/L9r9AL5O"], 270]]]```

Version 12.1 also has what one can think of as 3D generalizations of these kinds of textures:

 ✕ `Graphics3D[Style[Icosahedron[], HatchShading[]]]`

It looks pretty good even in black and white:

 ✕ `Graphics3D[Style[Icosahedron[], HatchShading[]], Lighting -> "Accent"]`

 ✕ ```Plot3D[Exp[-(x^2 + y^2)], {x, -2, 2}, {y, -2, 2}, PlotStyle -> {White, StippleShading[]}, Mesh -> None, Lighting -> "Accent"]```

### Advancing the Computational Aesthetics of Visualization (December 2020)

One of our objectives in Wolfram Language is to have visualizations that just “automatically look good”—because they’ve got algorithms and heuristics that effectively implement good computational aesthetics. In Version 12.2 we’ve tuned up the computational aesthetics for a variety of types of visualization. For example, in 12.1 this is what a SliceVectorPlot3D looked like by default:

 ✕ `SliceVectorPlot3D[{y + x, z, -y}, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}]`

Now it looks like this:

Since Version 10, we’ve also been making increasing use of our PlotTheme option, to “bank switch” detailed options to make visualizations that are suitable for different purposes, and meet different aesthetic goals. So for example in Version 12.2 we’ve added plot themes to GeoRegionValuePlot. Here’s an example of the default (which has been updated, by the way):

 ✕ `GeoRegionValuePlot[CloudGet["https://wolfr.am/ROWDoxAw"] -> "GDP"]`

And here it is with the "Marketing" plot theme:

 ✕ ```GeoRegionValuePlot[CloudGet["https://wolfr.am/ROWDoxAw"] -> "GDP", PlotTheme -> "Marketing"]```

Another thing in Version 12.2 is the addition of new primitives and new “raw material” for creating aesthetic visual effects. In Version 12.1 we introduced things like HatchFilling for cross-hatching. In Version 12.2 we now also have LinearGradientFilling:

 ✕ ```Graphics[Style[Disk[], LinearGradientFilling[{RGBColor[1., 0.71, 0.75], RGBColor[0.64, Rational[182, 255], Rational[244, 255]]}]]]```

And we can now add this kind of effect to the filling in a plot:

 ✕ ```Plot[2 Sin[x] + x, {x, 0, 15}, FillingStyle -> LinearGradientFilling[{RGBColor[0.64, Rational[182, 255], Rational[244, 255]], RGBColor[1., 0.71, 0.75]}, Top], Filling -> Bottom]```

To be even more stylish, one can plot random points using the new ConicGradientFilling:

 ✕ ```Graphics[Table[ Style[Disk[RandomReal[20, 2]], ConicGradientFilling[RandomColor[3]]], 100]]```

### Golden Knots, and Other Material Matters (May 2021)

We’ve talked about it almost since Version 1.0. And now finally in Version 12.3 it’s here! Realistic rendering of surfaces as materials. Here’s a knot, rendered as if it’s made of gold:

 ✕ ```Graphics3D[{MaterialShading["Gold"], KnotData["SolomonSeal", "ImageData"]}]```

Here’s a surface, in velvet:

 ✕ ```Plot3D[Sin[x + y^2], {x, -7, 7}, {y, -2, 2}, PlotStyle -> MaterialShading["Velvet"]]```

In Version 12.3 we’re supporting a bit more than a dozen standard named materials (with more to come). But MaterialShading is also set up to allow you to specify in detail explicit physical and geometrical properties of materials—so you can get effects like this:

 ✕ ```Graphics3D[{MaterialShading[<|"BaseColor" -> Texture[\!\(\* GraphicsBox[ TagBox[RasterBox[CompressedData[" 1:eJzsvceSJEuSJNi7e9nj/sL+xVz3OEQ7NDPVVfXqocyMDODIMMbYMUaBEryq 7un/XGbRqKb9gsiLEXk6eVqYqamyKBGbmIiw/N+fzT89/O//8i//4v6f+PrT Tfj/OM5N/K//F/7zF8MdDYz7u/9qePeDe+e/fP4/cPBf/jf8w/n8HVeNn+V2 HJphYEeRmyRhUeKH5jqa5+i+O3HtsWtZYWAGoRVFXpracWQEHo64SezEsZdl OGJHofqv4Xt2GPhZ5kSJnxQ4gjMxiJPwTDdNcQuMY4ehl6VOHPlpGuQZ75sV fpq7aWKGnhF6YVlYGDNLcEcc9HJ+476YFQaM8jKrp0FeeGkm941wIzdLvSwJ yzKu6qRu4roJq4rXphkmP7EdO4hxSZBjvdHE5TiYlRWGOsYMAleWZkZYVwJM LD9244zjpwnw8TNOAOf4eR6VFb7VwnXfMQLXS/O0apOySeo2bmoPl0ShLYD4 eYYp4XI1goOh8ixrurKbF+2c+Oc9/j3+Pf49/u+OfzdLmpZLDvx8Oi26uUNs /YljAf8HUxvZBn4PLA2GUAayBTHgEFVlMZ2ZoY8bWVFgCf74YCjNs3EEIAR5 ScOF/sg2u9W2Wa513wuLIihygOAXWB3mgzFDHMnaLmlbbgbeJVwdzmnTWnHI kblw2CUA+BPX0l0HN8WikroDwgJynLZdOVs4nAOskAY8XsJ2nlgcwwLSICuw u3AcN4WB1E7D9LA6rAhn4l4mtxYhwglBUWAoLir0y9k876ZYPv6K2/myH2j6 JCb+WYL5590sbbqorrAc7J9iNgtKmonbTMbBsD6txr2H8QG+4J/2+Pf49/j3 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Lighting -> "ThreePoint"]```

Notice, by the way, the new "ThreePoint" setting for Lighting—essentially a simulation of a standard placement of lights in a photography studio.

The story of modeling light interacting with surfaces—and “physically based rendering”—is a complicated one, that Version 12.3 has a whole monograph about:

### Lighting Goes Symbolic (December 2021)

Lighting is a crucial element in the perception of 3D graphics. We’ve had the basic option Lighting for specifying overall lighting in 3D scenes ever since Version 1.0. But in Version 13.0 we’re making it possible to have much finer control of lighting—which has become particularly important now that we support material, surface and shading properties for 3D objects.

The key idea is to make the representation of light sources symbolic. So, for example, this represents a configuration of light sources

 ✕

which can immediately be used with the existing Lighting option:

 ✕

But the new possibility is to “separately light” different objects in a scene, by specifying different symbolic “lighting styles” for them:

 ✕

By the way, another new feature in Version 13.0 is the built-in Torus primitive:

 ✕

## New Graphics & Visualization

### Look at It Now: HiDPI (March 2020)

Back in 1988 when we released Version 1.0 a typical computer display was maybe 640 pixels across (oh, and it was a CRT). And I was recently using some notebooks of mine from the 1990s (yes, they still work, which is spectacular!), and I was amazed at what a small window size they were made for. But as I write this today, I’m looking at two 3000-pixel displays. And 4k displays aren’t uncommon. So one of the things we’ve done for Version 12.1 is to add systemwide support for the new world of very-high-resolution displays.

One might think that would be easy, and would just “come with the operating system”. But actually it’s taken two years of hard work to deliver full HiDPI support. Well over a thousand carefully designed icons and other assets went from being bitmaps to being work-at-any-size algorithmic graphics. Everything about rasterization (not just for Rasterize, but for 3D graphics textures, etc. etc.) had to be redone. Sizes of things—and their interactions with the tower of kludges that operating systems have introduced over the years—had to be respecified and rethought.

But now it’s done. And we’re ready for displays of any resolution:

By the way, talking of displays, another “infrastructure” enhancement in Version 12.1 is moving to Metal and Direct3D 11 for 3D graphics rendering on macOS and Windows. Right now these just make 3D graphics modestly faster. But they also lay the groundwork for full multithreaded rendering, as well as VR, AR and more.

### 3D Array Plots (December 2020)

Back in 1984 I used a Cray supercomputer to make 3D pictures of 2D cellular automata evolving in time (yes, captured on 35 mm slides):

I’ve been waiting for 36 years to have a really streamlined way to reproduce these. And now finally in Version 12.2 we have it: ArrayPlot3D. Already in 2012 we introduced Image3D to represent and display 3D images composed of 3D voxels with specified colors and opacities. But its emphasis is on “radiology-style” work, in which there’s a certain assumption of continuity between voxels. And if you’ve really got a discrete array of discrete data (as in cellular automata) that won’t lead to crisp results.

And here it is, for a slightly more elaborate case of a 3D cellular automaton:

 ✕ ```Table[ArrayPlot3D[ CellularAutomaton[{14, {2, 1}, {1, 1, 1}}, {{{{1}}}, 0}, {{{t}}}], PlotTheme -> "Web"], {t, 10, 40, 10}]```

Another new ArrayPlot-family function in 12.2 is ComplexArrayPlot, here applied to an array of values from Newton’s method:

 ✕ ```ComplexArrayPlot[ Table[Nest[# - (#^3 - 1)/(3 #^2) &, x + I y, 30], {x, -2, 2, .01}, {y, -2, 2, .01}], ColorFunction -> None]```

### Can You Make a 10-Dimensional Plot? (December 2020)

It’s straightforward to plot data that involves one, two or three dimensions. For a few dimensions above that, you can use colors or other styling. But by the time you’re dealing with ten dimensions, that breaks down. And if you’ve got a lot of data in 10D, for example, then you’re probably going to have to use something like DimensionReduce to try to tease out “interesting features”.

But if you’re just dealing with a few “data points”, there are other ways to visualize things like 10-dimensional data. And in Version 12.2 we’re introducing several functions for doing this.

As a first example, let’s look at ParallelAxisPlot. The idea here is that every “dimension” is plotted on a “separate axis”. For a single point it’s not that exciting:

 ✕ ```ParallelAxisPlot[{{10, 17, 19, 8, 7, 5, 17, 4, 8, 2}}, PlotRange -> {0, 20}]```

Here’s what happens if we plot three random “10D data points”:

 ✕ `ParallelAxisPlot[RandomInteger[20, {3, 10}], PlotRange -> {0, 20}]`

But one of the important features of ParallelAxisPlot is that by default it automatically determines the scale on each axis, so there’s no need for the axes to be representing similar kinds of things. So, for example, here are 7 completely different quantities plotted for all the chemical elements:

 ✕ ```ParallelAxisPlot[ EntityValue[ "Element", {EntityProperty["Element", "AtomicMass"], EntityProperty["Element", "AtomicRadius"], EntityProperty["Element", "BoilingPoint"], EntityProperty["Element", "ElectricalConductivity"], EntityProperty["Element", "MeltingPoint"], EntityProperty["Element", "NeutronCrossSection"], EntityProperty["Element", "ThermalConductivity"]}]]```

Different kinds of high-dimensional data do best on different kinds of plots. Another new type of plot in Version 12.2 is RadialAxisPlot. (This type of plot also goes by names like radar plot, spider plot and star plot.)

RadialAxisPlot plots each dimension in a different direction:

 ✕ ```RadialAxisPlot[ EntityValue[ "Element", {EntityProperty["Element", "AtomicMass"], EntityProperty["Element", "AtomicRadius"], EntityProperty["Element", "BoilingPoint"], EntityProperty["Element", "ElectricalConductivity"], EntityProperty["Element", "MeltingPoint"], EntityProperty["Element", "NeutronCrossSection"], EntityProperty["Element", "ThermalConductivity"]}]]```

It’s typically most informative when there aren’t too many data points:

 ✕ ```RadialAxisPlot[ EntityValue[{Entity["City", {"Chicago", "Illinois", "UnitedStates"}], Entity["City", {"Dallas", "Texas", "UnitedStates"}], Entity["City", {"NewYork", "NewYork", "UnitedStates"}], Entity["City", {"LosAngeles", "California", "UnitedStates"}]}, {EntityProperty["City", "MedianHomeSalePrice"], EntityProperty["City", "TotalSalesTaxRate"], EntityProperty["City", "MedianHouseholdIncome"], EntityProperty["City", "Population"], EntityProperty["City", "Area"]}, "EntityAssociation"], PlotLegends -> Automatic]```

### Just Draw Anything (December 2020)

Type Canvas[] and you’ll get a blank canvas to draw whatever you want:

 ✕ `Canvas[]`

We’ve worked hard to make the drawing tools as ergonomic as possible.

Applying Normal gives you graphics that you can then use or manipulate:

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 ✕ ```GraphicsGrid[ Partition[ Table[Rasterize[Rotate[Normal[%], \[Theta]], ImageSize -> 50], {\[Theta], 0, 2 Pi, .4}], UpTo[8]], ImageSize -> 500]```

When you create a canvas, it can have any graphic as initial content—and it can have any background you want:

 ✕ ```Canvas[Graphics[ Style[Disk[], Opacity[.4, Red], EdgeForm[{Thick, Red}]]], Background -> GeoGraphics[ Entity["MannedSpaceMission", "Apollo16"][ EntityProperty["MannedSpaceMission", "LandingPosition"]]]]```

On the subject of drawing anything, Version 12.2 has another new function: MoleculeDraw, for drawing (or editing) molecules. Start with the symbolic representation of a molecule:

 ✕ `Molecule[Entity["Chemical", "Caffeine"]]`

Now use MoleculeDraw to bring up the interactive molecule drawing environment, make an edit, and return the result:

It’s another molecule now:

### Yet More Visualization (May 2021)

The Wolfram Language has a huge collection of built-in visualization functions—but there always seem to be more that we figure out can be added. We’ve had ListPlot3D since Version 1.0 (1988); we added ListPointPlot3D in Version 6.0 (2007)—and now in Version 12.3 we’re adding ListLinePlot3D.

Here’s a 3D random walk rendered with ListLinePlot3D:

 ✕ ```ListLinePlot3D[ AnglePath3D[RandomReal[{0 \[Degree], 20 \[Degree]}, {1000, 3}]]]```

If you give multiple lists of data, ListLinePlot3D plots them each separately:

 ✕ ```ListLinePlot3D[Table[Array[BitXor[#, n] &, 100], {n, 8}], Filling -> Axis]```

We first introduced plotting of vector fields in Version 7.0 (2008), with functions like VectorPlot and StreamPlot—that were substantially enhanced in Versions 12.1 and 12.2. In Version 12.3 we’re now adding StreamPlot3D (as well as ListStreamPlot3D). Here’s a plot of streamlines for a 3D vector field, colored by field strength:

 ✕ `StreamPlot3D[{y + x, y - x, z}, {x, -2, 2}, {y, -2, 2}, {z, -3, 3}]`

It’s one thing to make a plot; it’s another to give it axes. There’s a remarkable amount of subtlety in specifying how axes—and their ticks and labels—should be drawn. This is going to be a longer story, but in Version 12.3 we have the beginnings of symbolic axis specifications.

Axes work a bit like arrows: first you give the “core structure”, then you say how to annotate it. Here’s an axis that is linearly labeled from 0 to 100 on a line:

 ✕ `Graphics[AxisObject[Line[{{-1, -1}, {1, 1}}], {0, 100}]]`

And here’s a spiral axis (AKA labeling on a parametric curve):

 ✕ ```Graphics[ AxisObject[ Line[Table[{t Cos[t], t Sin[t]}, {t, 0, 5 Pi, 0.1}]], {0, 100}]]```

And, yes, it works in more ornate cases as well:

 ✕ `Graphics[AxisObject[HilbertCurve[3], {0, 100}, TickPositions -> 50]]`

Among many subtle issues, there’s the question of how tick labels should be oriented with respect to the axis:

 ✕ ```Graphics[ AxisObject[ Line[Table[{t Cos[t], t Sin[t]}, {t, 0, 5 Pi, 0.1}]], {0, 100}, TickPositions -> 50, TickLabelOrientation -> "Parallel"]]```

Talking of subtleties in graphics, here’s another one that’s addressed in Version 12.3. Say you’re making a dashed line:

 ✕ ```Graphics[{Dashing[{.2, .15}], Thickness[.05], Line[{{0, 0}, {1, .1}}]}]```

The numbers inside Dashing indicate the length of each dash, and between dashes. In Version 12.3 there’s an additional number you can give: the offset of the start of the first dash:

 ✕ ```Graphics[{Dashing[{.2, .15}, 0.1], Thickness[.05], Line[{{0, 0}, {1, .1}}]}]```

And something else is that you can control the “caps” on each dash, here making them rounded:

 ✕ ```Graphics[{Dashing[{.2, .15}, 0.1, "Round"], Thickness[.05], Line[{{0, 0}, {1, .1}}]}]```

These may all seem like micro-details—but they’re the kinds of things that are important in having Wolfram Language graphics look really good. Like, for example, if you have two dashed axes that cross, you probably want the “dashings” to line up:

 ✕ ```Graphics[{{Dashing[{0.1`, 0.05`}, 0.0195`], Line[{{-1, 0}, {1, 0}}]}, {Dashing[{0.1`, 0.05`}, 0.0195`], Line[{{0, -1}, {0, 1}}]}}, ImageSize -> 200]```

### Dates, and Infinities, in Plot Scales (December 2021)

In Version 13.0, the “coordinates” in plots don’t just have to be numbers; they can be dates too. So for example this means that all the usual plotting functions “just work” on things like time series:

 ✕

And there’s nothing stopping you having dates on multiple axes. Here’s an example of plotting time of day (a TimeObject) against date, in this case for email timestamps stored in a Databin:

 ✕

Another thing that’s new with axes—or rather with scales—in Version 13.0 is the ability to have infinite plot ranges:

 ✕

The way this works is that there’s a scaling function that maps the infinite interval to a finite one. You can use this explicitly with ScalingFunctions:

 ✕

Here’s a slightly more elaborate example, that includes a doubly infinite interval:

 ✕

### New Visualization Types (December 2021)

We’re constantly adding new types of built-in visualizations—not least to support new types of functionality. So, for example, in Version 13.0 we’re adding vector displacement plots to support our new capabilities in solid mechanics:

 ✕

Or in 3D:

 ✕

The plot shows how a given region is deformed by a certain displacement field. VectorPoints lets you include the displacement vectors as well:

 ✕

In Version 12.3 we introduced the function GeoGraphPlot for plotting graphs whose vertices are geo positions. In Version 13.0 we’re adding GeoGraphValuePlot which also allows you to visualize “values” on edges of the graph:

 ✕