U.U.D.M. Project Report 2020:53
       Measure theory, fractal geometry and 
       their applications on digital sundials
       Arvid Törnblom
       Examensarbete i matematik, 15 hp
       Handledare: Konstantinos Tsougkas
       Examinator: Veronica Crispin Quinonez
       Augusti 2020
                                        Department of Mathematics
                                               Uppsala University
          Abstract
          Thedigital sundial is a recent invention that displays the time of day in digits
          on a flat surface by projecting a shadow. It contains no moving or electrical
          parts and is based on a theorem from fractal geometry. In this thesis we
          will study this theorem which the sundial is based upon and the necessary
          measure theoretic background to understand the theorem.
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                         Contents
                         1 Introduction                                                                     1
                         2 Measure Theory                                                                   2
                             2.1   σ - algebras and Measures . . . . . . . . . . . . . . . . . . . .         2
                             2.2   Lebesgue Measure . . . . . . . . . . . . . . . . . . . . . . . . .        4
                             2.3   Non-measurable sets . . . . . . . . . . . . . . . . . . . . . . .         8
                         3 Fractal Geometry                                                                11
                             3.1   Fractals   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  11
                             3.2   Fractal Dimension . . . . . . . . . . . . . . . . . . . . . . . . .     17
                             3.3   Fractal Projection . . . . . . . . . . . . . . . . . . . . . . . . .    23
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