An approach to modeling heterogeneous objects as multidimensional point sets with multiple attributes (hypervolumes) is presented. Attributes given at each point represent object properties of arbitrary nature (material, physical, etc.). A proposed theoretical framework is based on a hybrid model of geometry and attributes combining a cellular representation and a functionally based constructive representation of dimensionally non-homogeneous entities. Hypervolume model components such as objects, operations and relations are introduced and outlined. We present examples of modeling a multi-layer geological structure with cavities and wells, time-dependent adaptive mesh generation, and conversion of a 3D implicit complex to the cellular representation.