Repeated Mediants Lead to Iterative Brackets
This Demonstration shows initial steps in the iterative bracket method for creating the gold list of increasingly close rational approximants of a given real number target.
The initial bracket is the pair of fractions and with smallest possible denominators such that both approximate the target and lie to its left and right. The number line shows the sequence of repeated mediants of the bracket elements, monotonically approaching the target from the side, before crossing over to the side at the mediant. All repeated mediants between the and the are guaranteed to be in the gold list: the repeated mediants most closely bracketing the target become the bracket elements for the next iteration step.
For comparison, you can choose to see the continued fraction representation of the target.
R. Lewis Caviness and Kenneth E. Caviness, "Repeated Mediants Lead to Iterative Brackets", from Wolfram Demonstrations Project, http://demonstrations.wolfram.com/RepeatedMediantsLeadToIterativeBrackets/, December 18, 2020.