Higher-dimensional Euclidean space

Many of you are probably asking what is outside the universe The answer is unclear. I must reiterate that this question supposes that the ultimate physical reality must be a Euclidean space of some dimension. That is, it presumes that if space is a hypersphere, then the hypersphere must sit in a four-dimensional Euclidean space, allowing us to view it from the outside. As the authors of the Scientific American article point out, nature need not adhere to this notion. It would be perfectly acceptable for the universe to be a hypersphere and not be embedded in any higher-dimensional space. We have difficulty visualizing this because we are used to viewing shapes from the outside. But there need not be an outside. ... 

As it has been shown in Ref [72], the fractional exponent v coincides with the fractal dimension of Cantor s set and indicates a fraction of the system states, being preserved during the entire evolution time t. Let us remind that Cantor s set is considered in one-dimensional Euclidean space (d=l) and therefore its fractal dimension d < by virtue of the fractal definition [86]. For fractal objects in Euclidean spaces with higher dimensions d> ) as V one should accept fractional part or [76, 77] ... 

The Euclidean distance mostly used does not consider correlations between the features. Points x, with a constant Euclidean distance from a fixed point m are situated on a circle (on a hypersphere in a higher dimensional feature space) around this point. In contrast, the Mahalanobis distance does consider the spatial distribution of the data points. For easier comparison the Euclidean and the Mahalanobis distances are given in the same notation in equations (9) and (lO), respectively. 

Consider now the same arrangement of A and A embedded in En+1, by regarding E" as a subspace of En+L A two-dimensional rotation in En+1 is defined by its (n-l)-dimensional axis and by the angle a of rotation in the remaining two dimensions. [Note that in a k-dimensional space, the axis of rotation is (k-2)-dimensional.] Choose the rotation axis in En+ as the (n-l)-dimensional subset defined as the reflection hyperplane E"- of condition x i = 0 in E". With respect to this axis, a rotation of angle a = 7C in the two-dimensional plane spanned by coordinates (xi, x +i) superimposes A on A in (n+l)-dimensions. Consequently, the object A is achiral in (n+l)-dimensions (i.e., when embedded in space E"+ ). Furthermore, the superimposition of mirror images performed in En+1 is a possible motion in any Euclidean space En+k (> of which En+ is a subspace, hence A is achiral in any higher dimensions. Consequently, chirality may occur only in the lowe.st dimension where A is embeddable. Q.E.D. 

---