I found that its length is neither changed nor unchanged if an endpoint of a line segment is cut off and hypothesized that a line is not composed of points in primary school. Similar findings and hypotheses about plane and space were obtained in middle and high school. An example of Russell's paradox with positive integers - ∅Pan -which I wrote out in the first year of university solved these problems unexpectedly. ∅Pan is a set group that all its sets are mutually included. By marking each set in ∅Pan with line, plane and space, I got ∅line, ∅plane, and ∅space, respectively. These geometric set groups have remarkable properties that lines are mutually included and mutually iterated in ∅line, and the same as planes in ∅plane and spaces in ∅space. Visualization models for ∅line through cross-composing timelines, for ∅plane through cross-monitoring screens, and for ∅space through cross-sharing cyberspaces were finally established when I was pursuing my doctoral degree. These results demonstrate that sets can mutually include and geometries can mutually iterate. I anticipate the present paper to be a starting point for a novel kind of set group and algebraic geometry.