I shall present the complex surface that I have named the spidron on account of its S-shape and spider-web appearance, and which, in its various transformations, deformations and combinations, has yielded extremely interesting spatial forms and tessellations. The so-called spidron nests, composed of 4, 6, 8 (generally an even number) of spidron arms, form reliefs that can be deformed in an accordion-like fashion. Such nests cut several of the regular and semiregular solids into two mirror-symmetrical halves. With the assistance of professor Emil Molnár, we are attempting to determine and classify all the space-filling shapes using the "Dsymbol" named after the three mathematicians Dress, Delaney and Delone. The first examples of that endeavour are shown using the cube cut in half by a spidron nest. We have discovered several sub-species of the spidron family, constructed from a sequence of a single type (45°, 90°, 45°) or alternating sequences of two different types of isosceles triangles.