![]() There is no better motivation to investigate shadows of “the upper world” then Plato’s Allegory of the Cave: “He will require to get accustomed to the sight of the upper world. Since we can properly visualize an object and its cutting hyperplane in the double orthogonal projection, we construct images of both-the object and its section. All these computer graphics methods use analytic representations and computations. A visualization of series of three-dimensional slices of time-varying data, considered to be a four-dimensional data field, is shown in Woodring et al. The author also gives extensive literature on methods of investigation of four-dimensional objects in computer vision. A similar process is applied to rotated positions of the polychoron for even better performance. The pentachoron is given by analytic coordinates of its vertices, while multiple slices with parallel hyperplanes are computed and visualized in perspective projection, and then, they are placed on a parabolic curve. A regular pentachoron is also the subject of investigation in Kageyama ( 2016). Several approaches of visualizing cuts of four-dimensional objects by 3-spaces have been developed in computer graphics. Ībbott in his Flatland Abbott ( 2006) describes an analogy when a 3-sphere appears as its circle sections in the flat two-dimensional world. Popular depiction of such net is Dalí’s Crucifixion (Corpus Hypercubus). Worth mentioning is also the investigation of polychora after decomposition into their three-dimensional net. Casas developed a polar perspective-method of visualization of the four-space created by continuous compositions of perspective projections onto spheres, which are later flattened. Interesting method of distorted visualization of four-dimensional objects on an example of tesseract is proposed in Casas ( 1984). A very good example with an extensive encyclopedia is Eusebeîa ( 2018). Nowadays, various models of polychora, in both parallel and central projections, may also be found all over the Internet. A famous treatise on regular polytopes as inductive dimensional analogies is in Coxeter ( 1973), Chapter VII. The properties of polychora are derived as natural analogies to the three-dimensional polyhedra. In contrast to parallel projection, in Séquin ( 2002) physical three-dimensional images of regular polychora in perspective projection are constructed. Orthogonal projections are also used in Miyazaki ( 1988), where a rolling four-dimensional die is visualized. It seems to be unbearable to carry three or more such images of complex objects, such as polychora, Footnote 1 in mind and perform operations with them. In both, an object is projected onto two-dimensional planes, and therefore we reconstruct the original image from three (by Lindgren and Slaby) or six (by Şerbǎnoius) planar images. The nearest synthetic methods of visualization of the 4-space with using descriptive geometry are to be found in Lindgren and Slaby ( 1968) and Şerbănoiu and Şerbănoiu ( 2017). To remind the method of double orthogonal projection, we start with a construction of a regular pentachoron. In four dimensions we proceed analogically and construct visualizations of solids, their sections and shadows, up to one small difference-we cannot really imagine them as a whole. However, if we also draw projections of intersections with planes or the shadow cast by the solid, we have much better understanding of its spatiality. Various projections give us some basic information about the solid. To comprehend spatial properties of a three-dimensional solid drawn on a two-dimensional paper, different strategies may be chosen. Specifically, we suggest the reader to go through the text with constructions and watch their corresponding interactive models in the supplementary material simultaneously. Even though we explain all the ideas in the text, descriptive geometry constructions are led in technical language, which is hard to be followed without understanding of processes and proper definitions in the given reference. We follow the terminology and notation of Zamboj ( 2018), where the introduction into a synthetic method of projection of four-dimensional objects onto two perpendicular 3-spaces and further bibliography on the development of their visualization is given. Our goal is to develop instructive synthetic techniques with the use of descriptive geometry and interactive graphics to improve our spatial perception of four-dimensional objects. ![]() We introduce visual methods of examination of the four-dimensional space. ![]()
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