Torus

Take a circle of radius r whose centre lies at distance R from a fixed axis, with R greater than r, and rotate it about that axis. The surface swept out is a torus. Two numbers describe it completely, and the ratio between them decides whether it looks like a bicycle tyre, a bagel, or a doughnut with barely any hole at all.

Mathematically it matters chiefly for what it is not: a sphere. The two cannot be deformed into one another without cutting, because the torus has a hole and the sphere does not, the torus is a surface of genus one, and noticing that this is a real, unremovable difference rather than a matter of shape is one of the starting points of topology. On a sphere every closed loop can be shrunk to a point. On a torus there are loops running around the hole and loops running through it that can never be shrunk away.

The same object turns up in places that look unrelated. It is the shape of the space in games where walking off one edge of the screen returns you at the opposite edge. It is the natural home of any system with two independent cyclic variables, which is why it appears constantly in mechanics and in the study of dynamical systems, where motion on a torus is the standard picture of two coupled oscillations.

Its geometry was studied early. Perseus, a Greek geometer of around 150 BCE, investigated the curves produced by cutting a torus with planes parallel to its axis. These are the spiric sections, named from the Greek word for the ring, and the family includes the hippopede and the Cassini ovals. They are the toric counterpart of the conic sections, and they are considerably harder, which is roughly why they are less famous.

A good deal of what modern sacred geometry says about the torus rests on observations that are simply correct, and those are worth setting out first and separately.

Vortex rings. A smoke ring, the bubble ring blown by a dolphin, the ring rising from a volcanic vent and the ring at the head of a mushroom cloud are all the same object: a body of fluid rotating about a closed circular axis, carrying itself forward, remarkably stable and long-lived. The flow really is toroidal and it really is self-sustaining for as long as viscosity permits.

Magnetic fields. The field of a bar magnet, of a current loop, and of the Earth is a dipole field, and its field lines form nested closed loops sweeping out a toroidal region around the source. The Earth's magnetosphere has that shape before the solar wind stretches it into a tail. The torus is also the geometry chosen for magnetic confinement in fusion research: a tokamak is a toroidal chamber precisely because a field line inside a torus never runs out of chamber to travel along, which a straight tube cannot manage.

Circulating fluids. Hadley circulation in the tropical atmosphere is a toroidal overturning, and convection in a heated fluid organises into rolls of the same kind.

The general principle behind all of these is real and worth stating plainly: any circulating flow that has to return to its own source, in a system possessing an axis, tends toward a toroidal shape. That follows from continuity and from having nowhere else to go. It is a good, genuine result, and it needs nothing added to it.

From the 1990s the torus became a central image in popular sacred geometry and New Age cosmology. The claim in its strong form is that the torus is the fundamental shape of energy at every scale (atom, cell, human being, planet, galaxy, universe), all the same self-sustaining circulation through a central axis.

The parts of that which are true are set out above. What is added is not.

Atoms are not tori. Electrons occupy orbitals whose shapes follow from quantum mechanics, and none of the standard orbital shapes is a torus. Galaxies are discs with a central bulge and a surrounding halo; the material orbits, but it does not circulate back through a hole. The universe may or may not have non-trivial topology, a toroidal universe is a genuine possibility that cosmologists test against microwave background data, and the data so far do not support it while not entirely excluding it, but that is a technical claim about global shape, not about energy flowing through an opening.

The human torus field is the most repeated version, and here the base fact is real. The heart generates a magnetic field detectable outside the body, in the range of tens to hundreds of picotesla at the chest surface, and measuring it is a working clinical technique called magnetocardiography, used among other things to localise ischaemic regions of the heart muscle.

The extensions are not established. The claim that this field extends around three feet in all directions and carries emotional information between people originates with the HeartMath Institute, is published largely outside mainstream peer review, and is not accepted in cardiology or biophysics. The heart's field is one bioelectric signal among many, is comparable in magnitude to skeletal muscle activity during movement, and falls off steeply with distance.

The torus is one of the most important shapes in mathematics and one of the commonest in fluid and electromagnetic behaviour. It does not need the additions.