Fehrian Branches or: How I Learned to Reduce y = x^2 + 1 and Love Mathematica

Visual schematic representations have built concrete intuition and aid in intuitive comprehension. Despite the enormous progress that modern abstract geometry has made since the late nineteenth century, the property of concentric circlesówhere they touch each other at the imaginary circular points at infinityóis still described as STRANGE in geometry books published in the twenty-first century. Some mathematicians may argue, "Since equations prove a theorem is correct, we do not mind if the visuals are incomplete or unavailable in the first place." However, a tradition dating back to the Greek era dictates that we describe the existence of a mathematical object geometrically and only consider it a mathematical reality when a purely geometric interpretation becomes possible. Even today, intuitive ways to grasp various facts and problems in mathematics are becoming increasingly vital. Presenting imaginary and infinitely distant elements using Mathematica's advanced pattern matching and rewriting capabilities should contribute to a correct understanding of complex geometry for many people, not just specialists.