Part 1: The Foundation of Existence. Chapter 4. The 3 Sphere as the Shape of the Universe

I think our universe has a shape that our everyday intuition does not really know. We are used to lines, planes, and spheres that you can still, in your mind, turn around and examine. A table is flat, a ball is round, the Earth is a kind of large marble that you can fly around. But as soon as you take one step further, to the question of how the universe as a whole is put together, our intuitions get stuck. We try to cram something that includes everything into the same mental toolkit we use to describe a room or a football stadium. That works up to a point, but at some point you have to admit that reality is probably less flat than our imagination.

In my mind, on the largest scales the universe is not infinite and flat, it is finite and without an edge. It has no outside, no wall you could bump into, and yet its total “content” is not infinite. The shape that fits this best is what mathematicians call a three sphere. The name sounds more abstract than it is. You can build it up step by step. A circle is the boundary of a flat disk. That boundary is itself finite, you can walk all the way around it, and it has no beginning or end. That is one dimensional curvature. An ordinary sphere, the surface of the Earth for example, is the two dimensional analogue, you can travel forever without finding an edge, but the total surface area is still finite. A three sphere is the same idea, but one dimension higher. It is the three dimensional “boundary” of a four dimensional ball. We cannot see that directly, but we can understand the behaviour, there is no edge there either, and yet the total volume is finite.

Why do I find such a three sphere so attractive as the shape of the universe? One important argument is that it frees us from artificial outsides. In many images the universe is some kind of expanding ball inside an emptiness around it. You see pictures of a balloon being inflated, with dots for the galaxies. That is didactically useful, but philosophically sloppy. Because what is that empty space outside the balloon, and why should the universe itself have to be an “object in something else”? In a three sphere you do not have that problem. The space of the universe is the entire structure. There is no outside. You can travel in any direction, farther and farther, and you never hit an edge, but after a very long time you do end up near where you started. Just like on Earth, but in three directions at once.

A second argument is that it fits the idea of a zero point field that closes on itself. In the previous chapter I described the zero point field as the layer underneath everything, a field that needs no beginning and that is present everywhere. If you give such a field the room to form a stable phase, it makes sense that it chooses a configuration without loose ends. An infinite flat space is one possibility, but a finite, edgeless structure has a different kind of elegance. A three sphere is exactly such a self closing pattern. You can see it as an eigenmode of the zero point field, a resonance in which the field as a whole takes on a coherent form. Our universe then becomes a kind of standing wave in the zero point field, and that standing wave has the topology of a three sphere.

Now you can say, fine, but what do I notice as a human being? The point is that the shape of the universe is not only a mathematical footnote, it carries through into everything we measure. A finite but edgeless universe means, for example, that there is no preferred position. There is no center where it all began, and no edge where things stop. Everywhere is “the middle”. That fits much better with the idea that there is no privileged observer. We are not sitting in the center of an expanding explosion, we are simply sitting in one place in a closed structure. A universe with the shape of a three sphere respects that principle in a natural way.

Another effect has to do with how we understand expansion. In the standard story, the expansion of the universe is often pictured as galaxies flying away from one another in a flat space. In a three sphere the picture is more subtle. You can imagine space as the surface of a balloon, but in three dimensions. The balloon itself is the spatial structure, and the expansion is the inflating of the balloon. The galaxies are not inside the balloon, they are in the skin. When the three sphere grows, the distances between all points increase. There is no place where it really “expands into something else”, the scale factor of the three sphere simply changes. It is internal, not an explosion outward.

I find that intuitively more honest. The universe does not have to conquer space, it does not have to break through an external nothing. It changes scale within its own form. That fits a zero point field in which phases and scale relationships can shift without any external environment. You then get a picture in which expansion is simply a property of the phase the field is in. The three sphere becomes larger or smaller, but it remains itself. That also makes it easier to think about cycles. A three sphere can grow, slow down, perhaps shrink again, without ever needing an edge or an outside world.

And still our brain will resist. A sphere in three dimensions we can still imagine, a three sphere we cannot. We cannot “see” what it is like to live in a three dimensional space that is itself the boundary of a four dimensional ball. But we do not have to. You can learn a lot by thinking one dimension lower. Imagine beings that are two dimensional and live on the surface of the Earth. Their world has length and width, but no height. They cannot rise above their own surface. Ask yourself what they would see. Locally their world is almost flat. They can build roads, build houses, do geometry. If they travel far enough, they eventually return to their starting point. If they take the time, they notice that their parallels meet at the poles, that the total surface area is finite, and that there is no edge. If someone tells them they live on a sphere, at first they will experience that as abstract mathematics. And yet it is true.

We are in exactly the same position with respect to a three sphere. Locally our space looks flat. The geometry in the living room follows the rules of Euclid perfectly well. Even on the scale of the solar system you can manage with straight lines and circles. Only on enormous scales, in cosmic structures, does the global shape begin to matter. Small deviations in angles, in relationships between distance and scale, in ratios of volume, they reveal a slight curvature. These are not things you notice directly as a human being, but they show up in the way light from distant sources presents itself to us. In the angles between cosmic beacons, in the statistics of the cosmic microwave background, in the arrangement of large structures. The three sphere does not show itself as a photograph, it shows itself as a pattern in the numbers.

What I find interesting is that in such a universe our optical misinterpretations count twice. GLV already assumes that light paths wind through all the small irregularities in the mass field, and that we therefore systematically overestimate distances and masses. Add on top of that a universe that is itself slightly curved as a three sphere, and you see how easily we can draw the wrong conclusions. We project everything back onto a fictional flat, infinite space, because that is the grid we use to calculate. But reality does not have to care about that grid. The three sphere is like the real globe, our flat map is only an approximation, and it comes with distortions.

Another important point is the absence of an absolute reference point. In many classical stories there is, implicitly, the idea that there is still some kind of “real” scale. A real time, a real distance, one set of coordinates against which everything must be measured. In a universe that has the shape of a three sphere, and that arises from a zero point field, that idea is hard to maintain. Locally you can have standard rulers and clocks, but on the largest scale there is no chosen center, no cosmic zero point in space. The only global invariant is the structure itself. Relativity, which in physics already says that there is no privileged inertial frame, gets a geometric extension here, there is also no privileged “place in the universe” where you have to begin the story.

There is also something human tied to this. We are used to thinking of ourselves as central, even when we dress it up in modern language. Long ago the Earth sat in the middle of the universe. Then the Sun, then perhaps our Milky Way, then the local cluster. In every phase we still crawled back to the center. A three sphere universe forces us to let that go. There is no center to claim. Every point is equally far from all “edges”, because there are no edges. That is unsettling under the surface, but also liberating. We no longer have to figure out where the real middle is. It does not exist.

There is yet another reason why I find the three sphere shape logical in combination with the zero point field. A finite, edgeless structure is the natural candidate for a stable mode. In nature you often see that systems that are free to evolve choose configurations that are closed, symmetric, and without loose ends. A drop becomes round, a bubble becomes spherical, a vibrating string chooses long, clean standing waves. In that sense, a three sphere is simply the spatial analogue of such an eigenmode in distance and time. You can see the zero point field as a system that can vibrate in many possible configurations, and our universe as one of the forms in which that remains in equilibrium on the largest scale.

The three sphere also makes it easier, conceptually, to deal with repetition and cycle. In an infinite flat space, the thought of a “return” feels artificial. You go infinitely far and you never come back. In a three sphere there is no principled objection to repetition. Take a ray of light, let it run around the whole structure a few times, and you get echoes of distant events. We have to stay sober about that, because the real cosmos is not perfectly transparent and static, but the idea that trajectories can close fits better with a closed form than with an infinite, flat space.

I am not saying that tomorrow, from your living room, you can notice that you live in a three sphere. Our brain is not built to visualise four dimensional balls, and our daily experience is far too local. But if you accept the zero point field as a basis, a field without a beginning, then a finite, edgeless universe becomes a surprisingly natural next step. It is a phase of the field that closes neatly on itself, without an outside, without a center, without an edge. A configuration in which space, time, energy, and information organise collectively into what we call the universe.

In the rest of the book I use this three spherical structure as an undertone. When I talk about how light paths mislead us, it is not only about small bends caused by tiny lenses, it is also about the fact that we project everything back onto a flat map while in reality we live on a curved globe. When I write about the apparent acceleration of expansion, the global shape of space is part of the story. And when I claim that the universe is not what it seems, I also mean this, we live in a closed, coherent form that we only partly feel through the light cone that reaches us.

After the zero point field and the three sphere, the framework is ready, a reality with a basis that needs no beginning, and a universe that itself takes on a stable, but not trivial shape. In the next chapters I can zoom in on what that means for gravity, for light, for black holes, and ultimately for the whole discussion about dark matter and dark energy. But underneath everything this foundation remains, we live in a wave in a field, and that wave has the form of a three sphere, finite, edgeless, and far less flat than it seems at first glance.