  In the beginning of the 20th century, A. N. Whitehead and T. de Laguna
proposed a new theory of space, known as {\em region-based theory of space}.
They did not present their ideas in a detailed mathematical form.
  In 1997, P. Roeper has shown that the locally compact Hausdorff spaces
correspond bijectively (up to homeomorphism and isomorphism) to some
algebraical objects which represent correctly Whitehead's ideas of {\em region}
and {\em contact relation}, generalizing in this way a previous analogous
result of de Vries concerning compact Hausdorff spaces (note that even a
duality for the category of compact Hausdorff spaces and continuous maps was
constructed by de Vries). Recently, a duality for the category of locally
compact Hausdorff spaces and continuous maps, based on Roeper's results, was
obtained by G. Dimov (it extends de Vries' duality mentioned above). In this
paper, using the dualities obtained by de Vries and Dimov, we construct
directly (i.e. without the help of the corresponding topological spaces) the
dual objects of Euclidean spaces, spheres, tori and Tychonoff cubes; these
algebraical objects completely characterize the mentioned topological spaces.
Thus, a mathematical realization of the original philosophical ideas of
Whitehead and de Laguna about Euclidean spaces is obtained.
