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S TRAIGHTENING THE SQUARE r tr t ts t rrst r What if you live in a flat

  1. S TRAIGHTENING THE SQUARE ❈♦♥❢♦r♠❛❧✴❛✣♥❡ ❣❡♦♠❡tr②✱ ✢❛t ❝♦♥♥❡❝t✐♦♥s ❛♥❞ t❤❡ ❙❝❤✇❛r③✲❈❤r✐st♦✛❡❧ ❢♦r♠✉❧❛ What if you live in a flat world where size is not well-defined? ❆r♥❛✉❞ ❈❤ér✐t❛t ❈◆❘❙✱ ■♥st✐t✉t ❞❡ ▼❛t❤é♠❛t✐q✉❡s ❞❡ ❚♦✉❧♦✉s❡ ▼❛r✳ ✷✵✶✾ ❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✶ ✴ ✸✸

  2. Linear maps in the plane do not preserve circles ❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✷ ✴ ✸✸

  3. ❚❤❡♥ ✐s ✐✛ ✐ts ❞✐✛❡r❡♥t✐❛❧ ✐s ❛ s✐♠✐❧✐t✉❞❡ ❡✈❡r②✇❤❡r❡✳ ❆ ❤♦❧♦♠♦r♣❤✐❝ ❛♥❞ ✐♥❥❡❝t✐✈❡ ♠❛♣ ✐s ❝❛❧❧❡❞ ✳ ❖♥❡ ❛ ✇❛② t♦ ♠❡❛s✉r❡ ❤♦✇ ❢❛r ✐s ❢r♦♠ ❜❡✐♥❣ ❛ s✐♠✐❧✐t✉❞❡ ✐s t♦ ❝♦♥s✐❞❡r t❤❡ ❡❧❧✐♣s❡ ♦❜t❛✐♥❡❞ ❛s t❤❡ ✐♠❛❣❡ ♦r ♣r❡✲✐♠❛❣❡ ♦❢ ❛ ❝✐r❝❧❡ ❜② ❛♥❞ ♠❡❛s✉r❡ ✐ts ✢❛t♥❡ss ✭❛✳❦✳❛✳ r❛t✐♦✮ ❂✏♠❛❥♦r✴♠✐♥♦r ❛①✐s✑ ✶✳ ❚❤❡ ✐t ✐s ❛ ❝♦♠♣❧❡① ♥✉♠❜❡r t❤❛t ❡♥❝♦❞❡s ❢❛✐t❤❢✉❧❧② t❤❡ ✢❛t♥❡ss ❛♥❞ ♦r✐❡♥t❛t✐♦♥ ♦❢ t❤❡ ♣r❡✲✐♠❛❣❡ ♦❢ ❛ ❝✐r❝❧❡ ❜② ✳ ✕ ❝❛♥ ❜❡ ❛♥② ❝♦♠♣❧❡① ♥✉♠❜❡r ♦❢ ♠♦❞✉❧✉s ✶ ✕ ✵ ✐✛ ✐s ❛ s✐♠✐❧✐t✉❞❡ Beltrami derivative • � ❂ t❤❡ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❂ ❛ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡ ≡ � ✷ • f ✿ ❛ ♠❛♣ ❢r♦♠ ✭❛ ❞♦♠❛✐♥ ♦❢✮ t❤❡ ♣❧❛♥❡ t♦ ✭❛ ❞♦♠❛✐♥ ♦❢✮ t❤❡ ♣❧❛♥❡✳ ■♥ t❤✐s ✇❤♦❧❡ t❛❧❦✱ ✇❡ ❛ss✉♠❡ f ♦r✐❡♥t❛t✐♦♥ ♣r❡s❡r✈✐♥❣✱ ✐✳❡✳ det( df ) > ✵✳ ❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✸ ✴ ✸✸

  4. ❖♥❡ ❛ ✇❛② t♦ ♠❡❛s✉r❡ ❤♦✇ ❢❛r ✐s ❢r♦♠ ❜❡✐♥❣ ❛ s✐♠✐❧✐t✉❞❡ ✐s t♦ ❝♦♥s✐❞❡r t❤❡ ❡❧❧✐♣s❡ ♦❜t❛✐♥❡❞ ❛s t❤❡ ✐♠❛❣❡ ♦r ♣r❡✲✐♠❛❣❡ ♦❢ ❛ ❝✐r❝❧❡ ❜② ❛♥❞ ♠❡❛s✉r❡ ✐ts ✢❛t♥❡ss ✭❛✳❦✳❛✳ r❛t✐♦✮ ❂✏♠❛❥♦r✴♠✐♥♦r ❛①✐s✑ ✶✳ ❚❤❡ ✐t ✐s ❛ ❝♦♠♣❧❡① ♥✉♠❜❡r t❤❛t ❡♥❝♦❞❡s ❢❛✐t❤❢✉❧❧② t❤❡ ✢❛t♥❡ss ❛♥❞ ♦r✐❡♥t❛t✐♦♥ ♦❢ t❤❡ ♣r❡✲✐♠❛❣❡ ♦❢ ❛ ❝✐r❝❧❡ ❜② ✳ ✕ ❝❛♥ ❜❡ ❛♥② ❝♦♠♣❧❡① ♥✉♠❜❡r ♦❢ ♠♦❞✉❧✉s ✶ ✕ ✵ ✐✛ ✐s ❛ s✐♠✐❧✐t✉❞❡ Beltrami derivative • � ❂ t❤❡ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❂ ❛ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡ ≡ � ✷ • f ✿ ❛ ♠❛♣ ❢r♦♠ ✭❛ ❞♦♠❛✐♥ ♦❢✮ t❤❡ ♣❧❛♥❡ t♦ ✭❛ ❞♦♠❛✐♥ ♦❢✮ t❤❡ ♣❧❛♥❡✳ ■♥ t❤✐s ✇❤♦❧❡ t❛❧❦✱ ✇❡ ❛ss✉♠❡ f ♦r✐❡♥t❛t✐♦♥ ♣r❡s❡r✈✐♥❣✱ ✐✳❡✳ det( df ) > ✵✳ ❚❤❡♥ f ✐s holomorphic ✐✛ ✐ts ❞✐✛❡r❡♥t✐❛❧ df ✐s ❛ s✐♠✐❧✐t✉❞❡ ❡✈❡r②✇❤❡r❡✳ ❆ ❤♦❧♦♠♦r♣❤✐❝ ❛♥❞ ✐♥❥❡❝t✐✈❡ ♠❛♣ ✐s ❝❛❧❧❡❞ conformal ✳ ❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✸ ✴ ✸✸

  5. ❚❤❡ ✐t ✐s ❛ ❝♦♠♣❧❡① ♥✉♠❜❡r t❤❛t ❡♥❝♦❞❡s ❢❛✐t❤❢✉❧❧② t❤❡ ✢❛t♥❡ss ❛♥❞ ♦r✐❡♥t❛t✐♦♥ ♦❢ t❤❡ ♣r❡✲✐♠❛❣❡ ♦❢ ❛ ❝✐r❝❧❡ ❜② ✳ ✕ ❝❛♥ ❜❡ ❛♥② ❝♦♠♣❧❡① ♥✉♠❜❡r ♦❢ ♠♦❞✉❧✉s ✶ ✕ ✵ ✐✛ ✐s ❛ s✐♠✐❧✐t✉❞❡ Beltrami derivative • � ❂ t❤❡ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❂ ❛ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡ ≡ � ✷ • f ✿ ❛ ♠❛♣ ❢r♦♠ ✭❛ ❞♦♠❛✐♥ ♦❢✮ t❤❡ ♣❧❛♥❡ t♦ ✭❛ ❞♦♠❛✐♥ ♦❢✮ t❤❡ ♣❧❛♥❡✳ ■♥ t❤✐s ✇❤♦❧❡ t❛❧❦✱ ✇❡ ❛ss✉♠❡ f ♦r✐❡♥t❛t✐♦♥ ♣r❡s❡r✈✐♥❣✱ ✐✳❡✳ det( df ) > ✵✳ ❚❤❡♥ f ✐s holomorphic ✐✛ ✐ts ❞✐✛❡r❡♥t✐❛❧ df ✐s ❛ s✐♠✐❧✐t✉❞❡ ❡✈❡r②✇❤❡r❡✳ ❆ ❤♦❧♦♠♦r♣❤✐❝ ❛♥❞ ✐♥❥❡❝t✐✈❡ ♠❛♣ ✐s ❝❛❧❧❡❞ conformal ✳ ❖♥❡ ❛ ✇❛② t♦ ♠❡❛s✉r❡ ❤♦✇ ❢❛r df ✐s ❢r♦♠ ❜❡✐♥❣ ❛ s✐♠✐❧✐t✉❞❡ ✐s t♦ ❝♦♥s✐❞❡r t❤❡ ❡❧❧✐♣s❡ ♦❜t❛✐♥❡❞ ❛s t❤❡ ✐♠❛❣❡ ♦r ♣r❡✲✐♠❛❣❡ ♦❢ ❛ ❝✐r❝❧❡ ❜② df ❛♥❞ ♠❡❛s✉r❡ ✐ts ✢❛t♥❡ss ✭❛✳❦✳❛✳ r❛t✐♦✮ K ❂✏♠❛❥♦r✴♠✐♥♦r ❛①✐s✑ > ✶✳ ❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✸ ✴ ✸✸

  6. Beltrami derivative • � ❂ t❤❡ ❝♦♠♣❧❡① ♥✉♠❜❡rs ❂ ❛ ❊✉❝❧✐❞❡❛♥ ♣❧❛♥❡ ≡ � ✷ • f ✿ ❛ ♠❛♣ ❢r♦♠ ✭❛ ❞♦♠❛✐♥ ♦❢✮ t❤❡ ♣❧❛♥❡ t♦ ✭❛ ❞♦♠❛✐♥ ♦❢✮ t❤❡ ♣❧❛♥❡✳ ■♥ t❤✐s ✇❤♦❧❡ t❛❧❦✱ ✇❡ ❛ss✉♠❡ f ♦r✐❡♥t❛t✐♦♥ ♣r❡s❡r✈✐♥❣✱ ✐✳❡✳ det( df ) > ✵✳ ❚❤❡♥ f ✐s holomorphic ✐✛ ✐ts ❞✐✛❡r❡♥t✐❛❧ df ✐s ❛ s✐♠✐❧✐t✉❞❡ ❡✈❡r②✇❤❡r❡✳ ❆ ❤♦❧♦♠♦r♣❤✐❝ ❛♥❞ ✐♥❥❡❝t✐✈❡ ♠❛♣ ✐s ❝❛❧❧❡❞ conformal ✳ ❖♥❡ ❛ ✇❛② t♦ ♠❡❛s✉r❡ ❤♦✇ ❢❛r df ✐s ❢r♦♠ ❜❡✐♥❣ ❛ s✐♠✐❧✐t✉❞❡ ✐s t♦ ❝♦♥s✐❞❡r t❤❡ ❡❧❧✐♣s❡ ♦❜t❛✐♥❡❞ ❛s t❤❡ ✐♠❛❣❡ ♦r ♣r❡✲✐♠❛❣❡ ♦❢ ❛ ❝✐r❝❧❡ ❜② df ❛♥❞ ♠❡❛s✉r❡ ✐ts ✢❛t♥❡ss ✭❛✳❦✳❛✳ r❛t✐♦✮ K ❂✏♠❛❥♦r✴♠✐♥♦r ❛①✐s✑ > ✶✳ ❚❤❡ Beltrami derivative ✐t ✐s ❛ ❝♦♠♣❧❡① ♥✉♠❜❡r Bf ( z ) t❤❛t ❡♥❝♦❞❡s ❢❛✐t❤❢✉❧❧② t❤❡ ✢❛t♥❡ss ❛♥❞ ♦r✐❡♥t❛t✐♦♥ ♦❢ t❤❡ ♣r❡✲✐♠❛❣❡ ♦❢ ❛ ❝✐r❝❧❡ ❜② df ✳ ✕ Bf ( z ) ❝❛♥ ❜❡ ❛♥② ❝♦♠♣❧❡① ♥✉♠❜❡r ♦❢ ♠♦❞✉❧✉s < ✶ ✕ Bf = ✵ ✐✛ df ✐s ❛ s✐♠✐❧✐t✉❞❡ ❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✸ ✴ ✸✸

  7. ❯s❡s✱ ❡①✐st❡♥❝❡✱ ❢♦r♠✉❧❛❄ Ellipse fields ❆♥ ellipse field ✐s t❤❡ ❞❛t❛ ♦❢ ✭✐♥✜♥✐t❡s✐♠❛❧✮ ❡❧❧✐♣s❡s ❛tt❛❝❤❡❞ t♦ ❡❛❝❤ ♣♦✐♥t ♦❢ ❛ ❞♦♠❛✐♥✳ ❚❤❡✐r s✐③❡ ✐s ♥♦t r❡❧❡✈❛♥t✱ ♦♥❧② t❤❡✐r ❞✐r❡❝t✐♦♥ ❛♥❞ ✢❛t♥❡ss✱ ❡♥❝♦❞❡❞ ❜② ❛ ❝♦♠♣❧❡① ♥✉♠❜❡r µ ( z ) ✇✐t❤ t❤❡ s❛♠❡ ❝♦♥✈❡♥t✐♦♥ ❛s ❢♦r Bf ✳ Straightening t❤❡ ❡❧❧✐♣s❡ ✜❡❧❞ ✐s s♦❧✈✐♥❣ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ Bf = µ ✱ ✐✳❡✳ ✜♥❞✐♥❣ ❛ ❞❡❢♦r♠❛t✐♦♥ f ♦❢ t❤❡ ♣❧❛♥❡ ✇❤♦s❡ ❞✐✛❡r❡♥t✐❛❧ s❡♥❞s ❛❧❧ t❤❡ ❡❧❧✐♣s❡s t♦ ❝✐r❝❧❡s✳ ❆r♥❛✉❞ ❈❤ér✐t❛t ✭❈◆❘❙✱ ■▼❚✮ ❙tr❛✐❣❤t❡♥✐♥❣ t❤❡ sq✉❛r❡ ▼❛r✳ ✷✵✶✾ ✹ ✴ ✸✸

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