Received Month X, XXXX; revised Month X, XXXX; accepted Month X, XXXX; posted Month X, XXXX (Doc. ID XXXXX); published Month X, XXXX Metamaterials are being used to model various exotic "optical spaces" for such applications as novel lenses and cloaking. While most effort is directed towards engineering of continuously changing dielectric permittivity and magnetic permeability tensors, an alternative approach may be based on lattices of metamaterial waveguides. Here we demonstrate the power of the latter technique by presenting metamaterial lattice models of various 4D spaces.OCIS Codes: 160.3918 Electromagnetic metamaterials can be used to emulate highly unusual "optical spaces" enabling such applications as novel transformation optics (TO) based lenses and cloaking [1][2][3]. This development is enabled by the newfound freedom of continuous control of the local dielectric permittivity εik and magnetic permeability μik tensors in electromagnetic metamaterials. Even though less prominent, another direction in metamaterial research may be characterized as "lattice-based metamaterial models". These models are based on the networks of metamaterial waveguides, which control propagation of the electromagnetic signals within a given 3D volume. Examples of this approach may be found in couple of recent experimental realizations of electromagnetic cloaking [4,5]. The goal of our paper is to demonstrate that the latter approach to "optical space" design with metamaterials is also very powerful, and it may supplement the more common approach of continuous engineering of ε ι k and μik tensors. For example, lattice-based metamaterial designs allow straightforward realization of various non-Euclidean "optical spaces", which were suggested as a pathway to broadband cloaking [6]. We demonstrate the ultimate power of lattice-based approach by presenting metamaterial lattice models of various hypercubic 4D spaces, which are impossible to emulate using conventional transformation optics. While our claim may seem unusual, we must point out that "hypercubic network architectures" for multiprocessor computing are being widely described in computer engineering literature. It is well known that hypercubic lattice networks of any spatial dimension D can be projected onto 3D space. Moreover, many of these hypercubic networks had been realized in the experiment [7]. An obvious advantage of implementing a lattice-based 4D metamaterial "optical hyperspace" is that any 3D nonEuclidean space may easily fit into such 4D space [8]. Therefore, a natural platform for the proposed nonEuclidean broadband TO designs will be provided.The simplest model of a 4D lattice projected onto regular 3D space is presented in Fig.1, which shows a perspective projection of an elementary unit of a 4D hypercubic spatial lattice, and a perspective projection of 2x2x2x2 region of the hypercubic lattice onto a region of 3D space. The elementary unit is filled with metal, except for the black lines, which represent pieces of thin single mode coaxial waveguides, wh...