Is there such a thing as negative or complex dimensions?

Question

I was explaining to my young son what it means for something to be in 1D, 2D and 3D, casually implying that 0D is a point and that there could be higher dimensions.

The question popped in his little head as he listened: is there something that is in -1 dimension?

And now, I can't stop thinking about it... I know we consider fractals to be in rational dimensions, but has anyone ever thought what would be something in a negative amount of dimensions? What about imaginary dimensions?

Sorry about your brain, I know you're also trying to imagine that, now. But my kid needs an answer: are dimensions bounded to $\mathbb{Q}+$? Or is there some kind of fringe theory that accepts exotic objects in $3i-7$ dimensions?

https://math.stackexchange.com/q/3319950/42969, https://math.stackexchange.com/q/3766408/42969 — Martin R, Aug 13 '24 at 10:32
@J.G.: Just to be clear for others, transcendental dimension possibilities for a fractal dimension is actually a fairly ho-hum situation as compared to negative-valued dimensions or non-real-valued dimensions (these latter requiring different notions of dimension than any of the half-dozen commonly used fractal dimensions). In fact, the ordinary Cantor middle thirds set has a transcendental dimension. — Dave L. Renfro, Aug 13 '24 at 11:26
@DaveL.Renfro Absolutely right. I brought up transcendental dimensions because the OP described fractal dimensions as rational. — J.G., Aug 13 '24 at 11:49
Hausdorff-dimensions are non-integers , but any reasonable usage of dimensions involves positive real numbers. — Peter, Aug 13 '24 at 12:53
@Peter: involves positive real numbers -- I'd replace "positive" with "non-negative" (e.g. topological dimension zero and Hausdorff dimension zero and packing dimension zero and Minkowski dimension zero). (continued) — Dave L. Renfro, Aug 13 '24 at 14:16
Also, even just restricting ourselves to fractal dimensional notions, the dimension can be a gauge function (this was even done in Hausdorff's original 1919 paper) and dimension prints and other ideas, although for basic introductory notions, probably non-negative real numbers captures "reasonable usage". — Dave L. Renfro, Aug 13 '24 at 14:16

Is there such a thing as negative or complex dimensions?

0 Answers0