Pi is simply the ratio of the circumference of a circle to its diameter. Why it goes on without pattern and with no end is because, apparently, the ratio cannot be written as a fraction of two integers.
Your inability to conceive infinity is common. You state that everything MUST have a defined conclusion due to its being physical matter. Well, what if physical matter is infinite? Have you ever heard of the laws of thermodynamics?
You limit your mind by making unwarranted assumptions. Good luck with your infinite chase.
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