Can we visualize higher dimensions? After 40+ years working on the math, the best I can do is imagine lower-dimensional analogs and projections, like this hypercube. Some claim to truly see higher dimensions. I’m not one of them.

What if they require "senses" we don't yet understand?

Absolutely agree, Professor. Rather than trying to see higher dimensions, I model how they dynamically express themselves. In my framework, time has a fifth-dimensional hydrodynamic extension, its behaviour governs dilation, inertia, and gravity. It's not about visualization; it's about understanding how the universe moves.

It depends on what you mean by “visualize”. A lot of higher-dimensional things, you can’t exactly imagine, but you can certainly imagine imagining, in a way that’s not exactly not visual. For example, imagine the 8-dimensional Lie algebra su(3), the tangent space of the SU(3) manifold at the identity (up to a factor of i, depending on the convention). Now imagine that entire space undergoing exponentiation, curling up periodically, infinitely surjectively, into the compact manifold of SU(3). Thinking of the topology of SU(3) as a fiber bundle whose base space is the 5-sphere of constant-norm triplets, and fibers are the SU(2) stabilizer subgroup of each triplet, and thinking of su(3) as isomorphic to R8, you can imagine eight orthogonal dimensions of infinite extent, curling up into that beautifully non-abelian symmetry group at the heart of QCD. If I can imagine that, then surely you can as well, probably with much more clarity. When I imagine it, there is definitely a visual aspect to it. It feels like my visual cortex is getting involved computationally, somehow… it’s math, but it’s not just equations, it’s “seeing” it. Doesn’t it feel that way for you too? That there are forms of sight beyond the usual dimensional format of photons hitting a 2D array of rods and cones?

That's because you're trying to imagine them outside of you instead of inside of you

Ever tried shrooms?

If there are higher spatial dimensions, then we should record their interaction with ours, which is not the case experimentally. By pouring a bottle of ink on a stack of paper, we record traces in a lower spatial dimension.

It depends on what you mean by dimensions: simple, passive one-dimensional lines or the real ones in our universe?

Aristotle would argue that you cannot. Your imagination is constituted from your sense, so only if you could actually see higher dimensions could you actually imagine them. But since the former is impossible, so is the latter.

Can’t get there from here (our little/limited slice of existence). But that doesn’t “make it NOT so.” Just sayin’. 😊

Fascinating insight, Dr. Greene! Your book The Fabric of the Cosmos was my first deep dive into these mind-bending concepts. It’s amazing how much imagination and analogy help bridge the gap to higher dimensions—even if we can't truly "see" them.