An Isometric Embedding of the Impossible Triangle into the Euclidean Space of Lowest Dimension

Zhenbing Zeng, Yaochen Xu, Zhengfeng Yang, Zhi-Bin Li

Abstract: The impossible triangle, invented independently by Oscar Reutersvärd and Roger Penrose in 1934 and 1957, is a famous geometry configuration that cannot be realized in our living space. Many people admitted that this object could be constructed in the four-dimensional Euclidean space without rigorous proof. In this paper, we prove that the isometric embedding problem can be decided by finite points on the configuration, then applying Menger and Blumenthal’s classical method of Euclidean embedding of finite metric space we determined the lowest Euclidean dimension, and finally using Maple obtained the coordinates of the isometric embedding. Our investigation shows that the impossible triangle is impossible to be isometrically embedded in the dimension four Euclidean space, but there is an isometric embedding to the dimension five space.