Ask Ethan: Why are maps of the cosmos always oval-shaped?

The Universe is a vast and expansive place. From any location, you have total freedom to look in any direction you like: up or down, left or right, and near or far, to any distance in any direction that you choose. (Well, so long as there isn’t anything nearby in the way of a more distant object that you want to observe.) It’s like you have a buffet, an omnidirectional buffet, of targets to choose from. You can even imagine observing it all: not just the half of the sky you can see by lying down in a field on a clear night, but in all directions all at once, like if you had an array of lenses that looked around in all 360° at once (plus the ability to view 90° up and down from the horizontal), that gathered light from all possible angles simultaneously.

And yet, when we show images of the cosmic microwave background — whether from COBE, WMAP, Planck, or a different mission — they’re almost always shown as oval-shaped. What does that oval shape actually show us, and why do astronomers make that specific visualization choice? That’s what Ed Matzenik wants to know, writing in to ask:

“I don’t understand the projections we see of the CMB. They are usually a circle or an oval. Is that the whole sky or just a section? If I was looking at a sphere from inside I don’t know how I’d represent it on a flat sheet…hope you can clear up this mystery for me.”

Honestly, the first time I encountered them — and remember, I’m a professional cosmologist who first encountered them in graduate school — I suffered from almost exactly the same puzzlement. Let’s begin with something we’re much more familiar with in order to get started: planet Earth.

This view of the Earth comes to us courtesy of NASA’s MESSENGER spacecraft, which had to perform flybys of Earth and Venus in order to lose enough energy to reach its ultimate destination: Mercury. Several hundred images, taken with the wide-angle camera in MESSENGER’s Mercury Dual Imaging System (MDIS), were sequenced into a movie documenting the view from MESSENGER as it departed Earth. Earth, an oblate spheroid, rotates roughly once every 24 hours on its axis and moves through space in an elliptical orbit around our Sun.

This is going to sound obvious, but the first thing you have to realize about planet Earth is that, to a first approximation, its shape is spherical. The most accurate tool we use to model and represent planet Earth is a three-dimensional structure: a globe. A globe, after all, is spherical in shape, and when you see land masses and watery regions, the globe does an excellent job of displaying things accurately in a wide variety of ways.

In particular, there are four things that are kept, or preserved, when you represent a spherical structure onto a two-dimensional globe.

1. You preserve the perpendicularity of lines of latitude and lines of longitude. For all latitudes (from +90° at the north pole to -90° at the south pole) and all longitudes (from either -180° to +180° or from 0° to 360°, as you prefer), where latitude and longitude lines intersect, they make right angles to one another.
2. You preserve the accuracy of the surface area of every structure. Greenland, for example, appears to be only a small fraction of the size of South America, and Antarctica is larger than Australia but smaller than Africa.
3. You have a fully connected map, one with no gaps or discontinuities between adjacent regions, countries, or waterways.
4. And you have undistorted shapes, where the coastlines and boundaries of any feature are accurate depictions of their actual, physical shapes.

This projection shows the two hemispheres of the Earth at once on what’s known as a Bacon Globular projection. Earth is a globe, and displaying two side-by-side pictures of the “near side” and “far side” of the Earth together is one way to display this curved surface on a flat screen.

But if you attempt to display the Earth as a flat, two-dimensional map, rather than a map on a curved (in this case, spherical) surface, now you’ve run into a problem.

In fact, it’s the same problem you run into if you very, very carefully try to peel an orange without ripping or tearing the skin into multiple pieces, and then attempt to lay the orange peel down flat onto a table. You’ll discover, almost immediately, that you absolutely cannot lay the orange peel flat without the skin either distorting or tearing or both. In fact, the more you try to flatten the peel itself, the more likely the peel will start to rip apart and tear.

There’s an important reason for this: simple geometry. When you take a curved surface, like the surface of a sphere, you can do things like draw a geometrical shape — a regular polygon, if you like — onto the surface. What you can then do is measure the angles at each vertex. Now, if you compared the angles of that same shape that were drawn onto a flat surface, you’d find that they weren’t identical! For example, if you drew an equilateral triangle onto:

• a positively curved surface, like a sphere, you’d find that the sum of the three internal angles was greater than 180°,
• a flat surface, like a flat sheet of paper, you’d find that the sum of the three internal angles always equaled exactly 180°,
• and if you drew it on a negatively curved surface, like a Pringles potato chip or a horse’s saddle, you’d see the sum of the three internal angles was always less than 180°.

The angles of a triangle add up to different amounts depending on the spatial curvature present. A positively curved (top), negatively curved (middle), or flat (bottom) Universe will have the internal angles of a triangle sum up to more, less, or exactly equal to 180 degrees, respectively. Advances in non-Euclidean geometry preceded their application to physics.

So now let’s come back to the orange peel analogy: attempting to take a perfectly peeled orange and to lay the orange peel down flat onto a two-dimensional surface. Only, instead of using an orange peel, we’re going to imagine using something that has a far more interesting set of features to it: the surface of planet Earth.

You might object and say, “Hey, how can you ‘peel’ the surface of the Earth and lay it down flat?” That’s not a thing we’re physically capable of doing, so where is this argument even going?

The point is that we don’t need to do it physically; we don’t need a physical orange peel to perform this mapping. All we need to do is have an understanding of geometry and its limitations. The orange peel exercise is just an illustration; we can mathematically map any well-defined geometric surface onto any other surface. The catch is that if the surfaces are of a fundamentally different curvature from one another, the mapping won’t preserve everything. For example, if you take a physical sphere (or the surface of a physical sphere, i.e., a spherical shell) and try to lay it flat, the sphere is bound to tear, and you’re going to wind up with something like the Goode homolosine projection.

This projection of the spherical Earth onto a flat surface is known as the Goode homolosine projection, preserving areas and largely preserving the perpendicularity of latitude and longitude lines, but sacrificing connectedness and intuitiveness. If it looks like an orange peel stretched out onto a flat surface to you, you’re not alone.

This is, indeed, a “map of the world,” and we can extend it to produce a map of any spherical shell: including the night sky in any wavelength of light that we choose. It shows the full world, including all the continents and oceans. It has a lot of really wonderful properties that mapmakers adore.

• It’s an equal-area projection, which means that the area of every feature is preserved, and equivalent to a representation of the actual area of the space it maps itself.
• It accurately projects the spatial distribution of phenomena, as well as the distances between those phenomena.
• And it has very minimal shape distortion, as the shapes of the land masses on this map are very close to the actual shapes of the continents in three-dimensional space.

However, it has some drawbacks, too. The most egregious is that this is not a simply-connected map: it has “gaps” in it that aren’t really there. For example, Antarctica isn’t broken up into four chunks, but rather is one contiguous land (and ice) mass. For another, lines of latitude and longitude aren’t all perpendicular to one another, but make severely distorted angles at the highest latitudes, leading to shape distortion in those places.

So why did we develop this projection? It was to provide an alternative to the most common type of projection used in cartography: the Mercator projection.

This map shows the world on a Mercator projection. While the perpendicularity of latitude and longitude lines are preserved and the map is fully connected, the drawback is that shapes and areas are distorted at high latitudes. In fact, this map doesn’t show the highest or lowest five degrees of latitude, as they would be stretched even more severely than the components of the Earth displayed here.

If you’ve seen a map of the world — such as from your elementary school classroom — there’s a very good chance this was the one you were looking at. The Mercator projection has two big advantages over the Goode homolosine projection.

• First, it’s simply connected, in the sense that it displays no gaps between adjacent features. You can move east, west, north, or south at any point along your map and simply go to the next feature that’s adjacent to it. That’s a big advantage.
• And second, lines of latitude and longitude are always perpendicular to one another. This is true even at high latitudes.

But there are downsides, too. The biggest downside is that area is not preserved in the Mercator projection, and this becomes obvious when you look at the size of Antarctica (which is not larger than Asia), or when you compare the sizes of Greenland and South America (which are not comparable, as South America is around eight times as large as Greenland). Areas are distorted to unphysically large sizes at high latitudes, with the greatest distortions of areas (and shapes) coming near the poles.

In fact, the shortcomings of the Mercator projection, particularly in its misrepresentation of angular size and on account of its latitude-dependent accuracies, led to a remarkable new projection that changed the game: the Mollweide projection.

This map shows the entire globe of the Earth projected onto a Mollweide projection, where areas are accurate and well-preserved and the map is fully connected, with no gaps. However, the perpendicularity of latitude and longitude is sacrificed at high latitudes and far away from the centrally projected longitude. The areas of the land masses and oceans, however, are accurate.

Above, you see planet Earth projected Mollweide-style. This combines two of the big advantages of the Goode homolosine (orange-peel-like) projection and the Mercator projection together: it’s an equal-area projection, where areas are preserved accurately at all latitudes and longitudes, while at the same time being a simply but fully connected map of a sphere. It even preserves the perpendicularity of latitude and longitude lines in some cases: at equatorial latitudes (i.e., a latitude of 0°) and at one particular longitude (e.g., at 0° longitude along the prime meridian, as shown here, although that’s arbitrary). You can see the true size of Antarctica and Greenland here, and recognize just how severely the Mercator projection is compared to this one.

But there are drawbacks to the Mollweide projection as well. Sure, there are cartographic drawbacks — the fact that latitude and longitude lines are not perpendicular and the fact that shapes, particularly at high latitudes, are distorted as well — but there’s another drawback that mapmakers won’t talk to you about: it’s not immediately intuitive the way a Mercator projection is. Because our planet’s surface is locally flat (i.e., when you walk outside, you can’t readily detect the curvature of the Earth), we’re used to thinking of the Earth in terms where you can move north, south, east, or west just as easily: there is no “better” direction at any point. But the Mollweide projection favors central latitudes and longitudes when it comes to more accurate shapes. Overall, it’s not an intuitive, familiar way to view an entire sphere all at once.

The most comprehensive view of the cosmic microwave background, which is the oldest light observable in the Universe, shows us a snapshot of what the cosmos was like just 380,000 years after the onset of the hot Big Bang. This projection of the entire sky is an example of a Mollweide projection: preserving areas over the entire sky but at the expense of perpendicular latitude/longitude lines.

Now, let’s come to the sky. Whether we’re looking at the cosmic microwave background (i.e., at the sky in microwave wavelengths of light), a map of the stars and/or the gas within the Milky Way (i.e., at the sky in optical or near-infrared light), or at the extragalactic sources that are present all throughout the cosmos (i.e., at the sky with the Milky Way, or even the whole Local Group, subtracted out), the fact is that we’re looking “out” at a full sphere, which projects to a spherical shell from our perspective.

That brings us to the crux of this week’s Ask Ethan question: how can we make sense of these oval-shaped maps of the Universe?

The biggest thing you need to do to “make sense” of it is to recognize that we are showing the entire sky, all 360° around in longitude and all latitudes as well, from +90° to -90°, projected onto a single oval, where the height of the oval is exactly half of the width of the oval. (As from +90° to -90° in latitude is half the number of degrees of the full 360° available in longitude.) It is the full sky, but projected onto one oval-shaped map. Areas (or angular sizes) are preserved on this map, and it is a simply, fully connected map as well. The thing we sacrifice is areas and undistorted shapes of features, as we lose both of these at high latitudes and at longitudes far away from the centrally displayed one.

Gaia’s all-sky view of our Milky Way Galaxy and neighboring galaxies. The maps show the total brightness and color of stars (top), the total density of stars (middle), and the interstellar dust that fills the galaxy (bottom). Note how, on average, there are approximately ~10 million stars in each square degree, but that some regions, like the galactic plane or the galactic center, have stellar densities well above the overall average. All three of these maps are shown as Mollweide projections of the entire sky.

Of course, there are many, many other projections one can choose; it truly is arbitrary. But different projections all have their own drawbacks. Some other possible choices include:

• The HEALPix projection, which maintains latitude and longitude perpendicularity near the equator but sacrifices connectedness (to preserve area) near the poles.
• The Eckert IV projection, which is kind of a compromise between Mollweide and Mercator, distorting shapes severely but preserving areas, while sacrificing perpendicularity, albeit in a less severe fashion than Mollweide.
• The Boggs eumorphic projection, which alleges (by the name eumorphic) to preserve the “good shapes” of land masses, sacrificing connectedness in a similar fashion to the orange-peel (or Goode homolosine) projection.
• And the van der Grinten projection, which is a fascinating but disastrous-at-the-poles projection that neither preserves areas nor perpendicularity of latitude and longitude. It preserves the Mercator projection’s shapes while reducing its distortions, but makes several other sacrifices in the process.

The big thing you should take away from this, however, is that when we in astronomy provide these oval-shaped maps of the Universe, we’re doing it Mollweide style: preserving areas and showing the full, fully connected sky, but doing so in a way that we only rarely view the Earth.

This unfamiliar view of the cosmic microwave background uses the familiar data from Planck, but projects it the same way most people project the Earth: onto a Mercator projection, which preserves the perpendicularity of latitude and longitude lines at the expense of distorting the areas and shapes of features at high latitudes. You can see the “stretchiness” of features near the top and bottom of the image as compared with the center.

We can do it in other fashions! Above is what we’d see if we projected the sky Mercator-style. There’s nothing stopping us from doing this; it’s a more “familiar” style of viewing a three-dimensional sphere projected onto a two-dimensional surface. Only, when it comes to something like a map of the cosmic microwave background, or a map of the Milky Way, or a map of the extragalactic night sky, the distorted areas at different latitudes give us a distorted view of what features in the Universe look like. We’ve decided, for good reasons (that become clear when you look at a Mollweide vs. Mercator projection of the sky), that preserving areas is much more important than preserving perpendicularity of latitude and longitude lines.

However, you shouldn’t be fooled into thinking that these maps actually represent what the Universe truly “looks like,” as they’re decomposing a sphere, where we can look out in all directions, and projecting it onto a flat, oval-shaped map that preserves the connectedness and area of the section of space we’re examining at the expense of perpendicularity and undistorted shapes. There is no way to preserve it all; you must sacrifice one or more of these traits in order to make a projection. You don’t need to be happy with the choice of a Mollweide projection either; it certainly has some undesirable attributes to it. The only problem is that all projections have undesirable attributes. When it comes to mapping the Universe, we’ve just collectively decided that the Mollweide projection is the “least bad” option out there.

Send in your Ask Ethan questions to startswithabang at gmail dot com!