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Are there any non self intersecting space filling curves?
There exist non-self-intersecting curves of nonzero area, the Osgood curves, but they are not space-filling. For the classic Peano and Hilbert space-filling curves, where two subcurves intersect (in the technical sense), there is self-contact without self-crossing.
Can a smaller cube fill a larger cube?
The sum of the volumes of the small cubes equals the volume of the large cube. This is ok for small numbers of cubes but we need to support this test of cubes with dimensions greater than 2^32.
Why are space filling curves called Peano curves?
In mathematical analysis, a space-filling curve is a curve whose range contains the entire 2-dimensional unit square (or more generally an n -dimensional unit hypercube ). Because Giuseppe Peano (1858–1932) was the first to discover one, space-filling curves in the 2-dimensional plane are sometimes called Peano…
Are there better algorithms for filling large cubes?
Even at 2^16 the number of small cubes required to fill the large cube is large enough that step 2 takes a while (O (n^2) checking each cube intersects no other). Is there a better algorithm? There seems to be some confusion over this.
How many iterations of the Hilbert curve are there?
Six iterations of the Hilbert curve construction, whose limiting space-filling curve was devised by mathematician David Hilbert.
Who was the first person to discover a space filling curve?
Because Giuseppe Peano (1858–1932) was the first to discover one, space-filling curves in the 2-dimensional plane are sometimes called Peano curves, but that phrase also refers to the Peano curve, the specific example of a space-filling curve found by Peano.
Where does the range of a space filling curve lie?
In the most general form, the range of such a function may lie in an arbitrary topological space, but in the most commonly studied cases, the range will lie in a Euclidean space such as the 2-dimensional plane (a planar curve) or the 3-dimensional space ( space curve ).