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Abstract
An arc or line segment in a Euclidean plane with the natural topology is selected. It is known the number of its endpoints is two and we calculate its midpoint and equation. Two finite sequences of points from the arc are selected which exclude the midpoint and endpoints. We define two circles which intersect at the arc’s midpoint and so that their diameters are each half the length of the arc. A finite collection of circles tangent to the interior of the arc at points of the selected sequences is created. The collection of circles is contained in the interiors of the two circles taken previously. This is accomplished using the dichotomous properties of the linear lemma and the Jordan curve theorem. Next a relation is defined to create the coefficients of a binary number using base two expansion. Now it can be shown as well by selecting a binary number that a continuum can be created to represent it. Again we take an arc in a plane with an origin and two circles which intersect at its midpoint and each contain one other endpoint of the arc. Circles are selected in the interiors of the two circles based on the coefficients of the base two expansion of the binary number. These circles are each tangent to the interior of the arc and positioned by the dichotomous natures of the line containing the arc and the coefficients of the base two expansion of the chosen binary number. The union of these objects in the plane creates a continuum as a subset of harmonic cellular continuum called the binary continuum. 