Can you have non integer dimensions?
Example of non-integer dimensions. That is, while the Hausdorff dimension of a single point is zero, of a line segment is 1, of a square is 2, and of a cube is 3, for fractals such as this, the object can have a non-integer dimension.
Does dimension have to be integer?
The number of dimensions need not even be an integer, as in the case of fractals—patterns that look the same on all scales. Cantor Set : Take a line, chop out the middle third and repeat ad infinitum. The resulting fractal is larger than a solitary point but smaller than a continuous line.
Is spacetime a fractal?
These recent theories of quantum gravity describe a fractal structure for spacetime itself, and suggest that the dimensionality of space evolves with time. Specifically, they suggest that reality is 2D at the Planck scale, and that spacetime gradually becomes 4D at larger scales.
What are the three dimensions of spacetime?
A three dimensional universe is made up of three dimensions, width, breadth, and height.
What are non integer dimensions?
Mathematicians have studied objects with non-integer dimension since the early 20’th century. The dimension is defined as the limiting value s, where hs(X) jumps from 0 to infinity. A line segment X of length 1 in the plane can be covered with n intervals of length 1/n and hs,r(X) = n (1/ns).
Where can I find Hausdorff dimension?
The Hausdorff Dimension We consider N=rD, take the log of both sides, and get log(N) = D log(r). If we solve for D. D = log(N)/log(r) The point: examined this way, D need not be an integer, as it is in Euclidean geometry. It could be a fraction, as it is in fractal geometry.
Is spacetime fractal and quantum coherent in the golden mean?
It appears likely that spacetime is fractal and quantum coherent in the golden mean. Physically, it appears to be a quantum coherent universe consisting of an infinite diversity of autonomous agents all participating in co-creating organic, fractal spacetime by their multitudinous coupled cycles of activities.