Orthogonal projections of cubes and regular simplices into a plane

We show that for every \({k\ge 2}\) and \({n\ge k}\), there is an \({n}\)-dimensional unit cube in \({\mathbb{R}^n}\) which is mapped to a regular \({2k}\)-gon by an orthogonal projection in \({\mathbb{R}^n}\) onto a \({2}\)-dimensional subspace. Moreover, by increasing dimension \({n}\), arbitrary large regular \({2k}\)-gon can be obtained in such a way. On the other hand, for every \({m\ge 3}\) and \({n\ge m-1}\), there is an \({n}\)-dimensional regular simplex of unit edge in \({\mathbb{R}^n}\) which is mapped to a regular \({m}\)-gon by an orthogonal projection onto a plane. Moreover, contrary to the cube case, arbitrary small regular \({m}\)-gon can be obtained in such a way, by increasing dimension \({n}\).