We have an anecdote about how he avenged himself on his neighbor, Zenon, by fashioning leather tubes that he ran to the joists of an upper room of Zenon's house, where Zeno used to entertain guests. We are told that Anthemius would, by running steam through the tube, create loud noises and vibration in he room, frightening the guests into thinking there was an earthquake. Also, he would flash incredibly bright light into Zenon's eyes with a concave mirror.
Possible? Well, he did write a treatise "On burning-glasses"; we don't have the treatise anymore, but enough of it existed in 1777 to be included in a work called Concerning wondrous machines by an L. Dupuy. He apparently studied and wrote on properties of mirrors and lenses, and supposedly described a camera obscura. He explained how to construct an ellipse using string, and he wrote a book on conic sections.
Anthemius' knowledge of conic sections and parabolas would have supported both his work on optics (known to the later "Second Ptolemy" Alhazen ibn al-Haytham) and his architectural aspirations when designing the dome of the new Hagia Sophia. He was able to create what is called a "pendentive": a design that allows a dome to be built onto a square base.
His success with Hagia Sophia led to him being also chosen—probably by Emperor Justinian—to design flood defenses at Dara in northern Mesopotamia.