❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛
❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ◆❛t✉r❡③❛
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛
▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ✲ P❘❖❋▼❆❚
❊st✉❞♦ ❙♦❜r❡ ♦ ❈á❧❝✉❧♦ ❞❡ ➪r❡❛s
❡ ❱♦❧✉♠❡s ❯t✐❧✐③❛♥❞♦ ♦ ▼ét♦❞♦ ❞❡
❊①❛✉stã♦ ❡ ♦ Pr✐♥❝í♣✐♦ ❞❡
❈❛✈❛❧✐❡r✐
†♣♦r
❋r❛♥❝✐s❝♦ ❞♦ ◆❛s❝✐♠❡♥t♦ ▲✐♠❛
s♦❜ ♦r✐❡♥t❛çã♦ ❞♦
Pr♦❢✳ ❉r✳ ▼❛♥❛ssés ❳❛✈✐❡r ❞❡ ❙♦✉③❛
❚r❛❜❛❧❤♦ ❞❡ ❝♦♥❝❧✉sã♦ ❞❡ ❝✉rs♦ ❛♣r❡s❡♥t❛❞♦ ❛♦ ❈♦r♣♦ ❉♦✲ ❝❡♥t❡ ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚ ❈❈❊◆✲❯❋P❇✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
◆♦✈❡♠❜r♦✴✷✵✶✸ ❏♦ã♦ P❡ss♦❛ ✲ P❇
L732e Lima, Francisco do Nascimento.
Estudo sobre o cálculo de áreas e volumes utilizando o Método de Exaustão e o Princípio de Cavalieri / Francisco do
Nascimento Lima.- João Pessoa, 2013.
66f. : Il.
Orientador: Manassés Xavier de Souza
Dissertação (Mestrado) – UFPB/CCEN
1. Matemática. 2. Áreas. 3. Volumes. 4. Método de
Exaustão. 5. Princípio de Cavalieri.
❆❣r❛❞❡❝✐♠❡♥t♦s
❆ ❉❡✉s✱ ♣♦r ❡st❛r s❡♠♣r❡ ♣r❡s❡♥t❡ ❡♠ ♠✐♥❤❛ ✈✐❞❛✦
❆ ❊❧✈í❞✐♦ ❋❡rr❡✐r❛ ❞❡ ▲✐♠❛✱ ♠❡✉ ♣❛✐✱ ♠❡s♠♦ ♥ã♦ ❡st❛♥❞♦ ♠❛✐s ❡♥tr❡ ♥ós✱ s✉❛ ❧❡♠❜r❛♥ç❛ ♠❡ ❞á ♠✉✐t❛ ❢♦rç❛❀ ❡ à ♠✐♥❤❛ ♠ã❡✱ ▼❛r✐❛ ■r✐s ❞♦ ◆❛s❝✐♠❡♥t♦ ▲✐♠❛✱ q✉❡ s❡♠♣r❡ ♠❡ ✐♥❝❡♥t✐✈♦✉ ❛♦ ❡st✉❞♦✳
❆♦s ♠❡✉s q✉❡r✐❞♦s ✜❧❤♦s✱ ■❛s♠✐♥ ●❛❜r✐❡❧❛ ❡ ❋r❛♥❝✐s❝♦ ●❛❜r✐❡❧✱ q✉❡ sã♦ ♦s ♠♦t✐✈♦s ❞❛ ♠✐♥❤❛ ❡①✐stê♥❝✐❛ ❡ ❞❡❞✐❝❛çã♦ ❛♦ tr❛❜❛❧❤♦✳
➚ ♠✐♥❤❛ ❡s♣♦s❛✱ ❈r✐st✐❛♥❡ ❈❛r✈❛❧❤♦✱ ♣❡❧♦ ❛♣♦✐♦ ♥♦s ♠♦♠❡♥t♦s ❞✐❢í❝❡✐s✳
❆♦s ♠❡✉s ✐r♠ã♦s✱ ■r✐♠❛r ▲✐♠❛ ❇❛t✐st❛✱ ▲✉✐③ ❆❧❜❡rt♦✱ ❖♠❛r ❞♦ ◆❛s❝✐♠❡♥t♦✱ ■r♠❛r ❞♦ ◆❛s❝✐♠❡♥t♦ ❡ ■r✐♥❡✐❞❡ ▲✐♠❛✱ q✉❡ s❡♠♣r❡ ♠❡ ❛♣♦✐❛r❛♠ ♥♦s ❡st✉❞♦s ❞❛ ▼❛t❡♠át✐❝❛✳
❆♦s ♣r♦❢❡ss♦r❡s ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛✱ ❞♦ ❞❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡✲ ♠át✐❝❛✱ q✉❡ ❧❡❝✐♦♥❛r❛♠ ♥♦ P❘❖❋▼❆❚✱ ♣♦✐s ❢♦r❛♠ ❡❧❡s q✉❡♠ ✐♥❝❡♥t✐✈❛r❛♠ ❛ ❜✉s❝❛ ❛♦s ❝❛❜❡❞❛✐s ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦✳
❆♦s ♠❡♠❜r♦s ❞❛ ❜❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛✱ ♣❡❧❛ ❞✐s♣♦s✐çã♦ ❡♠ ❛✈❛❧✐❛r ❡st❡ ❚r❛❜❛❧❤♦ ❞❡ ❈♦♥❝❧✉sã♦ ❞❡ ❈✉rs♦✱ ❡♠ ❡s♣❡❝✐❛❧ ❛♦ ♣r♦❢❡ss♦r ❉r✳ ▼❛♥❛ssés ❳❛✈✐❡r ❞❡ ❙♦✉③❛ q✉❡ ❝♦rr✐❣✐✉ ❡ ♦r✐❡♥t♦✉✱ ❝♦♠ ♠✉✐t❛ ❞❡❞✐❝❛çã♦✱ t♦❞♦ ♦ ♠❡✉ tr❛❜❛❧❤♦✳
❆♦s ❛♠✐❣♦s ❆❧②ss♦♥ ❊s♣❡❞✐t♦✱ ❆❧❞❡❝❦ ▼❡♥❡③❡s✱ ▼❛r❝❡❧♦ ❉❛♥t❛s✱ ❉②❡❣♦ ❆í❧❧♦ ❡ ❘♦♥❛❧❞♦ ♣❡❧❛s ✈❡r❞❛❞❡✐r❛s ❛♠✐③❛❞❡s ❝✉❧t✐✈❛❞❛s✱ ❣r❛ç❛s ❛♦ ❛❞✈❡♥t♦ ❞❛ ♠❛t❡♠át✐❝❛✱ ❡♠ ♥♦ss❛s ✈✐❞❛s✳
➚ ❙♦❝✐❡❞❛❞❡ ❇r❛s✐❧❡✐r❛ ❞❡ ▼❛t❡♠át✐❝❛ ✭❙❇▼✮ ♣❡❧❛ ❝r✐❛çã♦ ❞♦ ▼❡str❛❞♦ Pr♦✜s✲ s✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ✭P❘❖❋▼❆❚✮✱ ❞❛♥❞♦ ♦♣♦rt✉♥✐❞❛❞❡ ♣❛r❛ q✉❡ ♣r♦❢❡ss♦r❡s ❞❛ ❡❞✉❝❛çã♦ ❜ás✐❝❛ ♣♦ss❛♠ ♠❡❧❤♦r❛r ♦s s❡✉s ❝♦♥❤❡❝✐♠❡♥t♦s ♠❛t❡✲ ♠át✐❝♦s ❡ à ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛ ✭❯❋P❇✮ ♣♦r ❛❜r❛ç❛r ❡st❛ ✐❞❡✐❛✳
❉❡❞✐❝❛tór✐❛
❘❡s✉♠♦
❊st❡ tr❛❜❛❧❤♦ t❡✈❡ ❝♦♠♦ ♦❜❥❡t✐✈♦ ❞❡♠♦♥str❛r ❛❧❣✉♠❛s ❢ór♠✉❧❛s ♠❛t❡♠át✐❝❛s ✉s❛❞❛s ♥❛ ❣❡♦♠❡tr✐❛ ♣❛r❛ s❡r✈✐r ❞❡ ❢♦♥t❡ ❞❡ ❝♦♥s✉❧t❛ ♣❛r❛ ♣r♦❢❡ss♦r❡s ❡ ❛❧✉♥♦s ❞♦ ❊♥s✐♥♦ ❇ás✐❝♦✳ ■♥✐❝✐❛♠♦s ❝♦♠ ✉♠❛ ❞✐s❝✉ssã♦ ❞❛ ❤✐stór✐❛ ❞❛ ❣❡♦♠❡tr✐❛✱ s♦❜r❡ ♦s s❡✉s ♣r✐♠ór❞✐♦s ❡ ❛❧❣✉♥s ❞♦s ❣r❛♥❞❡s ♣❡♥s❛❞♦r❡s ❞❛ ●ré❝✐❛ ❆♥t✐❣❛✱ ❛ ✜♠ ❞❡ ❡♥t❡♥❞❡r♠♦s s❡✉ s✉r❣✐♠❡♥t♦✱ ❜❡♠ ❝♦♠♦ s❡✉ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦✳ P♦st❡r✐♦r♠❡♥t❡✱ ♠♦str❛♠♦s q✉❡ ❛ ♠❡❞✐❞❛ ❞❡ ✉♠ s❡❣♠❡♥t♦ é ✉♠ ♥ú♠❡r♦ r❡❛❧✱ q✉❡ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥✲ ❝✐❛ é ♣r♦♣♦r❝✐♦♥❛❧ ❛♦ s❡✉ r❛✐♦ ❡ q✉❡ ❛ ár❡❛ ❞♦ ❝ír❝✉❧♦ é ♣r♦♣♦r❝✐♦♥❛❧ ❛♦ q✉❛❞r❛❞♦ ❞♦ s❡✉ r❛✐♦✳ ❉❡♠♦♥str❛♠♦s q✉❡ ♣♦❧í❣♦♥♦s ❞❡ ár❡❛s ✐❣✉❛✐s ♣♦❞❡♠ s❡r ❡q✉✐❞❡❝♦♠♣♦♥í✲ ✈❡✐s ❡ ❝♦♥❝❧✉í♠♦s ♦❜t❡♥❞♦ ❛s ❢ór♠✉❧❛s ❞❡ ❝♦♠♦ ❝❛❧❝✉❧❛r ♦ ✈♦❧✉♠❡ ❞❡ ❛❧❣✉♥s só❧✐❞♦s ❣❡♦♠étr✐❝♦s ✉s❛♥❞♦ ♦ ♠ét♦❞♦ ❞❡ ❡①❛✉stã♦ ❡ ♦ ♣r✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐✳
P❛❧❛✈r❛s ❝❤❛✈❡s✿ ➪r❡❛s❀ ❱♦❧✉♠❡s❀ ▼ét♦❞♦ ❞❡ ❊①❛✉stã♦❀ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐✳
❆❜str❛❝t
❚❤✐s st✉❞② ❛✐♠❡❞ t♦ ❞❡♠♦♥str❛t❡ s♦♠❡ ♠❛t❤❡♠❛t✐❝❛❧ ❢♦r♠✉❧❛s ✉s❡❞ ✐♥ ❣❡♦♠❡tr② t♦ s❡r✈❡ ❛s ❛ r❡❢❡r❡♥❝❡ s♦✉r❝❡ ❢♦r t❡❛❝❤❡rs ❛♥❞ st✉❞❡♥ts ♦❢ ❇❛s✐❝ ❊❞✉❝❛t✐♦♥✳ ❲❡ ❤❛✈❡ ❜❡❣✉♥ ✇✐t❤ s♦♠❡ ♦❢ t❤❡ ❣r❡❛t t❤✐♥❦❡rs ♦❢ ❆♥❝✐❡♥t ●r❡❡❝❡ ❛♥❞ ✇✐t❤ ❛ ❞✐s❝✉ss✐♦♥ ♦❢ t❤❡ ❍✐st♦r② ♦❢ ●❡♦♠❡tr② s✐♥❝❡ ✐ts ✜rst ❞❛②s ✐♥ ♦r❞❡r t♦ ✉♥❞❡rst❛♥❞ ✐ts ❡♠❡r❣❡♥❝② ❛♥❞ ❞❡✈❡❧♦♣♠❡♥t✳ ❲❡✱ t❤❡♥✱ ❤❛✈❡ s❤♦✇❡❞ t❤❛t t❤❡ ♠❡❛s✉r❡ ♦❢ ❛ s❡❣♠❡♥t ✐s ❛ r❡❛❧ ♥✉♠❜❡r✱ t❤❛t t❤❡ ❧❡♥❣t❤ ♦❢ ❛ ❝✐r❝❧❡ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ ✐ts r❛❞✐✉s ❛♥❞ t❤❛t t❤❡ ❛r❡❛ ♦❢ ❛ ❝✐r❝❧❡ ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ sq✉❛r❡ ♦❢ ✐ts ♦✇♥ r❛❞✐✉s✳ ❲❡ ❛❧s♦ ❞❡♠♦♥str❛t❡❞ t❤❛t s❛♠❡✲❛r❡❛ ♣♦❧②❣♦♥s ❝❛♥ ❜❡ ❡q✉❛❧❧② ❞❡❝♦♠♣♦s❡❞✳ ■♥ t❤❡ ❡♥❞ ♦❢ t❤✐s st✉❞② ✇❡ ❞✐❞ ♦❜t❛✐♥ t❤❡ ❢♦r♠✉❧❛s ♦♥ ❤♦✇ t♦ ❝❛❧❝✉❧❛t❡ t❤❡ ✈♦❧✉♠❡ ♦❢ s♦♠❡ ❣❡♦♠❡tr✐❝ s♦❧✐❞s ✉s✐♥❣ t❤❡ ❡①❤❛✉st✐♦♥ ♠❡t❤♦❞ ❛♥❞ ❈❛✈❛❧✐❡r✐ ♣r✐♥❝✐♣❧❡✳
❑❡②✇♦r❞s✿ ❆r❡❛s❀ ❱♦❧✉♠❡s❀ ▼❡t❤♦❞ ♦❢ ❊①❤❛✉st✐♦♥❀ ❈❛✈❛❧✐❡r✐ ♣r✐♥❝✐♣❧❡✳
❙✉♠ár✐♦
▲✐st❛ ❞❡ ❋✐❣✉r❛s ✈✐✐✐
✶ ❯♠ P♦✉❝♦ ❞❡ ❍✐stór✐❛ ✶
✶✳✶ Pr✐♠ór❞✐♦s ❞❛ ●❡♦♠❡tr✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✷ ❆❧❣✉♥s ●r❛♥❞❡s P❡♥s❛❞♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷
✷ ❈♦♠♣r✐♠❡♥t♦✱ ➪r❡❛ ❡ ❱♦❧✉♠❡ ✻
✷✳✶ ❈♦♠♣r✐♠❡♥t♦ ❞❡ ✉♠ ❙❡❣♠❡♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✷✳✷ ➪r❡❛ ❞❡ ✉♠ ❘❡tâ♥❣✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✷✳✸ P♦❧í❣♦♥♦s ❊q✉✐❞❡❝♦♠♣♦♥í✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✷✳✹ ❈♦♠♣r✐♠❡♥t♦ ❞❡ ✉♠❛ ❈✐r❝✉♥❢❡rê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✺ ➪r❡❛ ❞❡ ✉♠ ❈ír❝✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✻ ❱♦❧✉♠❡ ❞❡ ❈♦r♣♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✻✳✶ ❈♦♥❝❡✐t♦ ❞❡ ❱♦❧✉♠❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✻✳✷ ❱♦❧✉♠❡ ❞❡ ✉♠ P❛r❛❧❡❧❡♣í♣❡❞♦ r❡tâ♥❣✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✻✳✸ ❱♦❧✉♠❡ ❞❡ ✉♠ P❛r❛❧❡❧❡♣í♣❡❞♦ ❖❜❧íq✉♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✻✳✹ ❱♦❧✉♠❡ ❞❡ ✉♠ Pr✐s♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼
✸ ❆❧❣✉♠❛s ❆♣❧✐❝❛çõ❡s ❞♦ ▼ét♦❞♦ ❞❡ ❊①❛✉stã♦ ✷✾
✸✳✶ ❱♦❧✉♠❡ ❞❡ ✉♠❛ P✐râ♠✐❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✸✳✷ ❱♦❧✉♠❡ ❞❡ ✉♠ ❈♦♥❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✸✳✸ ❱♦❧✉♠❡ ❞❡ ✉♠❛ ❊s❢❡r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻
✹ ❆❧❣✉♠❛s ❆♣❧✐❝❛çõ❡s ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ✸✽
✹✳✶ ❱♦❧✉♠❡ ❞❡ ✉♠ Pr✐s♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✹✳✷ ❱♦❧✉♠❡ ❞❡ ✉♠❛ P✐râ♠✐❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✹✳✸ ❱♦❧✉♠❡ ❞❡ ✉♠ ❈✐❧✐♥❞r♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻ ✹✳✹ ❱♦❧✉♠❡ ❞❡ ✉♠ ❈♦♥❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✹✳✺ ❱♦❧✉♠❡ ❞❡ ✉♠❛ ❊s❢❡r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✹✳✻ ➪r❡❛ ❞❡ ✉♠❛ ❊❧✐♣s❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶
❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✺✹
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✶✳✶ ❚❛❧❡s ❞❡ ▼✐❧❡t♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✷ ❊stát✉❛ ❞❡ ❆rq✉✐♠❡❞❡s ❞❡ ❙✐r❛❝✉s❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✷✳✶ ❙❡❣♠❡♥t♦ ❢♦r♠❛❞♦ ♣❡❧♦s ♣♦♥t♦s A✱ B ❡C✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼
✷✳✷ ❙❡❣♠❡♥t♦ AB ❢♦r♠❛❞♦ ♣♦rn s❡❣♠❡♥t♦s ❝♦♥❣r✉❡♥t❡s ❛u✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼
✷✳✸ ❘❡tâ♥❣✉❧♦ ❞❡ ❧❛❞♦s 3m ❡4m✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽
✷✳✹ ◗✉❛❞r❛❞♦ ❞❡ ❧❛❞♦ 1❞✐✈✐❞✐❞♦ ❡♠ N ♣❛rt❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾
✷✳✺ ❘❡tâ♥❣✉❧♦ ❞❡ ❧❛❞♦s ✐rr❛❝✐♦♥❛✐s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✷✳✻ P♦❧í❣♦♥♦s P1 ❡ P2✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶
✷✳✼ P♦❧í❣♦♥♦s ❡q✉✐❞❡❝♦♠♣♦♥í✈❡✐s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✷✳✽ ❚r✐â♥❣✉❧♦ ❡q✉✐❞❡❝♦♠♣♦♥í✈❡❧ ❝♦♠ ✉♠ r❡tâ♥❣✉❧♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✾ P♦❧í❣♦♥♦s ABCD ❡ ABEF✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸
✷✳✶✵ P❛r❛❧❡❧♦❣r❛♠♦ ABCD✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹
✷✳✶✶ P❛r❛❧❡❧♦❣r❛♠♦ EF P Q❡ ♦ ♣❛r❛❧❡❧♦❣r❛♠♦ EF GH✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹
✷✳✶✷ P♦❧í❣♦♥♦ q✉❛❧q✉❡r✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✶✸ P♦❧í❣♦♥♦P ❡ r❡tâ♥❣✉❧♦R✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺
✷✳✶✹ P♦❧í❣♦♥♦P′ ❡ r❡tâ♥❣✉❧♦ R′✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻
✷✳✶✺ ❚r✐â♥❣✉❧♦s s❡♠❡❧❤❛♥t❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✶✻ P♦❧í❣♦♥♦s ✐♥s❝r✐t♦ ❡ ❝✐r❝✉♥s❝r✐t♦ ♥❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✷✳✶✼ P♦❧í❣♦♥♦ ✐♥s❝r✐t♦ ❡ ♣♦❧í❣♦♥♦ ❝✐r❝✉♥s❝r✐t♦ ♥❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ λ✳ ✳ ✳ ✳ ✳ ✳ ✶✽
✷✳✶✽ P♦❧í❣♦♥♦ s❡♠❡❧❤❛♥t❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✶✾ P♦❧í❣♦♥♦ ✐♥s❝r✐t♦ ♥❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✷✵ ❚r✐â♥❣✉❧♦ ✐sós❝❡❧❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✷✶ ❯♠ ❝✉❜♦ ❡ ✉♠ r❡❝✐♣✐❡♥t❡ q✉❛❧q✉❡r✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✷✷ ❈✉❜♦ ✉♥✐tár✐♦ ❡ r❡tâ♥❣✉❧♦ ❞❡ ❧❛❞♦s ✐♥t❡✐r♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✷✸ ❯♠ ❝✉❜♦ ❡ ✉♠ ♣❛r❛❧❡❧❡♣í♣❡❞♦ r❡tâ♥❣✉❧♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✷✹ P❛r❛❧❡❧♦❣r❛♠♦ ♦❜❧íq✉♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✷✺ P❛r❛❧❡❧❡♣í♣❡❞♦ ♦❜❧íq✉♦ABCDA1B1C1D1✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻
✷✳✷✻ P❛r❛❧❡❧❡♣í♣❡❞♦ ♦❜❧íq✉♦ABCDA2B2C2D2✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼
✷✳✷✼ P❛r❛❧❡❧❡♣í♣❡❞♦ ♦❜❧íq✉♦ABB4A4A3B3C3D3✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼
✷✳✷✽ P❛r❛❧❡❧❡♣í♣❡❞♦ ❞❡ ❜❛s❡ tr✐❛♥❣✉❧❛r✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✷✾ Pr✐s♠❛ q✉❛❧q✉❡r✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✸✳✶ P✐râ♠✐❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✸✳✷ ❚r♦♥❝♦ ❞❛ ♣✐râ♠✐❞❡ ❡ ♣r✐s♠❛ ❞❡ ♠❡s♠❛ ❜❛s❡ ❡ ♠❡s♠❛ ❛❧t✉r❛ ✳ ✳ ✳ ✳ ✳ ✸✷ ✸✳✸ ❈♦♥❡ ❞❡ ❛❧t✉r❛H✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹
✸✳✹ ❈✐❧✐♥❞r♦ ❡ tr♦♥❝♦ ❞❡ ✉♠ ❝♦♥❡ ❞❡ ♠❡s♠❛ ❛❧t✉r❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✸✳✺ ❈✐❧✐♥❞r♦ ❞❡ ❛❧t✉r❛ H
n✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺
✸✳✻ ❙❡♠✐❡s❢❡r❛ ❞❡ r❛✐♦R✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻
✹✳✶ ❘❡s♠❛ ❞❡ ♣❛♣❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✹✳✷ ❋✐❣✉r❛s ❞❡ ár❡❛s ✐❣✉❛✐s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✹✳✸ ❙ó❧✐❞♦s ❞❡ ♠❡s♠♦ ✈♦❧✉♠❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✹✳✹ P❛r❛❧❡❧❡♣í♣❡❞♦ r❡tâ♥❣✉❧♦ ❡ ✉♠ ♣r✐s♠❛ ❞❡ ♠❡s♠❛ ár❡❛ ❞❛ ❜❛s❡✳ ✳ ✳ ✳ ✳ ✹✵ ✹✳✺ P✐râ♠✐❞❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✹✳✻ P✐râ♠✐❞❡ ❞❡ ❛❧t✉r❛ H✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷
✹✳✼ P✐râ♠✐❞❡s ❞❡ q✉❛❧q✉❡r✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✹✳✽ P✐râ♠✐❞❡s ❞❡ ♠❡s♠❛ ❜❛s❡ tr✐❛♥❣✉❧❛r ❡ ♠❡s♠❛ ❛❧t✉r❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✹✳✾ Pr✐s♠❛ ❞❡ ❜❛s❡ tr✐❛♥❣✉❧❛r✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✹✳✶✵ P✐râ♠✐❞❡s ❞❡ ✈♦❧✉♠❡s ✐❣✉❛✐s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✹✳✶✶ P♦❧í❣♦♥♦ ❞❡ ❜❛s❡ ♣♦❧✐❣♦♥❛❧ ❞❡ n ❧❛❞♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻
✹✳✶✷ Pr✐s♠❛ ❡ ❝✐❧✐♥❞r♦ ❞❡ ♠❡s♠❛ ár❡❛ ❞❛ ❜❛s❡ ❡ ♠❡s♠❛ ❛❧t✉r❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✹✳✶✸ ❈♦♥❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✹✳✶✹ ❚r✐â♥❣✉❧♦s s❡♠❡❧❤❛♥t❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✹✳✶✺ P✐râ♠✐❞❡ ❡ ❝♦♥❡ ❞❡ ♠❡s♠❛ ár❡❛ ❞❛ ❜❛s❡ ❡ ♠❡s♠❛ ❛❧t✉r❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✹✳✶✻ ❈✐❧✐♥❞r♦ ❡q✉✐❧át❡r♦ ❝✐r❝✉♥s❝r✐t❛ ❡♠ ✉♠❛ ❡s❢❡r❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✹✳✶✼ ❙ó❧✐❞♦ L à ❡sq✉❡r❞❛ ❡ só❧✐❞♦ L′ à ❞✐r❡✐t❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵
■♥tr♦❞✉çã♦
❆ ♥❡❝❡ss✐❞❛❞❡ ❞❡ s❛❜❡r s♦❜r❡ ✈♦❧✉♠❡s ❞❡ ❛❧❣✉♥s só❧✐❞♦s ❣❡♦♠étr✐❝♦s ✈❡♠ ❞❡s❞❡ ❛ ❛♥t✐❣✉✐❞❛❞❡✱ ❝♦♠ ❛s ♥❡❝❡ss✐❞❛❞❡s ♣rát✐❝❛s ❡ ♣❡❧❛s ❝✉r✐♦s✐❞❛❞❡s ❞♦ ❤♦♠❡♠ ❡♠ ❡♥❝♦♥tr❛r ♠ét♦❞♦s q✉❡ ❢❛❝✐❧✐t❛ss❡♠ ♦ s❡✉ ❝á❧❝✉❧♦✳
●r❛♥❞❡ ♣❛rt❡ ❞♦s ❧✐✈r♦s ❞♦ ❡♥s✐♥♦ ♠é❞✐♦ ♥ã♦ tr❛③ ❛s ❞❡♠♦♥str❛çõ❡s ❞❛s ❢ór♠✉❧❛s ❞❡ ❝♦♠♦ ❝❛❧❝✉❧❛r ✈♦❧✉♠❡s ❞♦s só❧✐❞♦s ❣❡♦♠étr✐❝♦s✱ ❝♦♠♦✿ ♣r✐s♠❛✱ ❝✐❧✐♥❞r♦✱ ♣✐râ♠✐❞❡✱ ❝♦♥❡ ❡ ❡s❢❡r❛❀ ❡♠ ❛❧❣✉♥s✱ ❛s ❞❡♠♦♥str❛çõ❡s ♥ã♦ ❡stã♦ tã♦ ❝❧❛r❛s ❡ ♣❡r❝❡❜❡♠♦s ❛ ❞✐✜❝✉❧❞❛❞❡ q✉❡ ❣r❛♥❞❡ ♣❛rt❡ ❞♦s ❛❧✉♥♦s ❛♣r❡s❡♥t❛ ♥♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞❛s ❢ór♠✉❧❛s✳
❆ ✐♠♣♦rtâ♥❝✐❛ ❞❡ss❛s ❞❡♠♦♥str❛çõ❡s ♣❛r❛ ❛ ✈✐❞❛ ❛❝❛❞ê♠✐❝❛ ❞♦s ❛❧✉♥♦s✱ ♣♦❞❡ s❡r ♦❜s❡r✈❛❞❛ ♥♦s P❛râ♠❡tr♦s ❈✉rr✐❝✉❧❛r❡s ◆❛❝✐♦♥❛✐s P❈◆✰✱ ✭✷✵✵✷✮✿
✧❯s❛r ❛s ❢♦r♠❛s ❣❡♦♠étr✐❝❛s ♣❛r❛ r❡♣r❡s❡♥t❛r ♦✉ ✈✐s✉❛❧✐③❛r ♣❛rt❡s ❞♦ ♠✉♥❞♦ r❡❛❧ é ✉♠❛ ❝❛♣❛❝✐❞❛❞❡ ✐♠♣♦rt❛♥t❡ ♣❛r❛ ❛ ❝♦♠♣r❡❡♥sã♦ ❡ ❝♦♥str✉çã♦ ❞❡ ♠♦❞❡❧♦s ♣❛r❛ r❡s♦❧✉çã♦ ❞❡ q✉❡stõ❡s ❞❛ ♠❛t❡♠á✲ t✐❝❛ ❡ ❞❡ ♦✉tr❛s ❞✐s❝✐♣❧✐♥❛s✳ ❈♦♠♦ ♣❛rt❡ ✐♥t❡❣r❛♥t❡ ❞❡st❡ t❡♠❛✱ ♦ ❛❧✉♥♦ ♣♦❞❡rá ❞❡s❡♥✈♦❧✈❡r ❤❛❜✐❧✐❞❛❞❡s ❞❡ ✈✐s✉❛❧✐③❛çã♦✱ ❞❡ ❞❡s❡♥❤♦✱ ❞❡ ❛r❣✉♠❡♥t❛çã♦ ❧ó❣✐❝❛ ❡ ❞❡ ❛♣❧✐❝❛çã♦ ♥❛ ❜✉s❝❛ ❞❡ s♦❧✉çã♦ ♣❛r❛ ♣r♦❜❧❡♠❛s✳✧✭♣✳ ✶✷✸✮✳ ✧❈♦♠♣r❡❡♥❞❡r ♦ s✐❣♥✐✜❝❛❞♦ ❞❡ ♣♦st✉❧❛❞♦ ❡ ♦✉ ❛①✐♦♠❛s ❡ t❡♦r❡♠❛s ❡ r❡❝♦♥❤❡❝❡r ♦ ✈❛❧♦r ❞❡ ❞❡♠♦♥str❛çõ❡s ♣❛r❛ ♣❡r❝❡❜❡r ❛ ▼❛t❡♠át✐❝❛ ❝♦♠♦ ❝✐ê♥❝✐❛ ❝♦♠ ❢♦r♠❛ ❡s♣❡❝í✜❝❛ ♣❛r❛ ✈❛❧✐❞❛r r❡s✉❧t❛❞♦s✧✭♣✳ ✶✷✺✮✳
❚❛♠❜é♠ ♥♦t❛♠♦s ✉♠❛ ✐♠♣♦rtâ♥❝✐❛ ❞❡ s❡ ❝♦♠♣r❡❡♥❞❡r ❡ss❡s ❝♦♥❝❡✐t♦s r❡❧❛❝✐♦✲ ♥❛❞♦s às ♥❡❝❡ss✐❞❛❞❡s ❞♦ ❝♦t✐❞✐❛♥♦✱ ♣♦✐s ♣❡r❝❡❜❡♠♦s ❡ss❛s ✜❣✉r❛s ❣❡♦♠étr✐❝❛s ❡♠ ✧♣❡ç❛s ♠❡❝â♥✐❝❛s✱ ❡♠❜❛❧❛❣❡♥s ❡ ❝♦♥str✉çõ❡s❀ ♣r♦❥❡çõ❡s✱ ♣❧❛♥✐✜❝❛çõ❡s✱ ❝♦rt❡s ❡ ❞❡s❡✲ ♥❤♦s✧✭P❈◆✰✱ ✷✵✵✷✱ ♣✳✶✷✺✮✳
P❡❧♦ ❢❛t♦ ❞❡ ♣❡r❝❡❜❡r♠♦s q✉❡ ♠✉✐t♦s ♣r♦❢❡ss♦r❡s ❞❛ ❊❞✉❝❛çã♦ ❇ás✐❝❛ ♥ã♦ ❞ã♦ ê♥❢❛s❡ ❛ ❡ss❡s ❝♦♥t❡ú❞♦s q✉❡ ❡♥✈♦❧✈❡♠ ❛s ✜❣✉r❛s ❣❡♦♠étr✐❝❛s ❡ ♣❡❧❛ ❢❛❧t❛ ❞❡ r✐❣♦r ❡♠ ❛❧❣✉♠❛s ❞❡♠♦♥str❛çõ❡s ❞❡ ❛❧❣✉♠❛s ❢ór♠✉❧❛s ❡♥❝♦♥tr❛❞❛s ♥♦s ❧✐✈r♦s ❞✐❞át✐❝♦s✱ ❢♦❝❛♠♦s ♥♦ss❛ ♣❡sq✉✐s❛ ♥❛s ❞❡♠♦str❛çõ❡s ❞❛s ❢ór♠✉❧❛s✱ s❡♠ ❢❛③❡r ✉s♦ ❞♦ ❝á❧❝✉❧♦ ✐♥t❡❣r❛❧✳
❉❡ss❛ ❢♦r♠❛ ♥♦ss♦ ♦❜❥❡t✐✈♦ é ♠♦str❛r ❝♦♠♦ ❡♥❝♦♥tr❛r ❛s ❢ór♠✉❧❛s ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❡ ár❡❛s ❡ ✈♦❧✉♠❡s ❞❡ ❛❧❣✉♥s só❧✐❞♦s ❣❡♦♠étr✐❝♦s ✉s❛♥❞♦ ♦ ▼ét♦❞♦ ❞❡ ❊①❛✉stã♦✱ ❝r✐❛❞♦ ♣♦r ❊✉❞ó①✐♦ ❡ ♠✉✐t♦ ✉s❛❞♦ ♣♦r ❆rq✉✐♠❡❞❡s ❞❡ ❙✐r❛❝✉s❛✱ ❡ ♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐✳
P❛r❛ ❡st❡ ❡st✉❞♦ ❞✐✈✐❞✐♠♦s ♥♦ss♦ tr❛❜❛❧❤♦ ❡♠ q✉❛tr♦ ♣❛rt❡s✿
◆♦ ❝❛♣ít✉❧♦ ✶✱ ❛♣r❡s❡♥t❛♠♦s ✉♠ ♣♦✉❝♦ ❞♦s ♣r✐♠ór❞✐♦s ❞❛ ❣❡♦♠❡tr✐❛✱ ❞♦ ❞❡✲
s❡♥✈♦❧✈✐♠❡♥t♦ ❡ ♦s ♣r✐♠❡✐r♦s ✐♥❞í❝✐♦s ❞❛ ❣❡♦♠❡tr✐❛ r✉❞✐♠❡♥t❛r✳ ❆❜♦r❞❛♠♦s s♦❜r❡ ♣❡♥s❛❞♦r❡s ❝♦♠♦ ❚❛❧❡s ❞❡ ▼✐❧❡t♦✱ ♦ q✉❛❧ ✐♥✐❝✐♦✉ ❝♦♠ ❛ ❣❡♦♠❡tr✐❛ ❞❡♠♦♥str❛t✐✈❛✱ ❡ P✐tá❣♦r❛s ❞❡ ❙❛♠♦s✱ q✉❡ ❢♦r♠♦✉ ❛ ❡s❝♦❧❛ ♣✐t❛❣ór✐❝❛✱ ❝r✐♦✉ ❛ ❚❡♦r✐❛ ❞♦s ◆ú♠❡r♦s ❡ ♦ ❢❛♠♦s♦ ❚❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s✱ q✉❡ ❞❡♠♦♥str❛ q✉❡ ❛ r❛✐③ ❞❡ ✷ ♥ã♦ é r❛❝✐♦♥❛❧✳ ❆❜♦r❞❛♠♦s t❛♠❜é♠ s♦❜r❡ ❊✉❝❧✐❞❡s ❞❡ ❆❧❡①❛♥❞r✐❛✱ ❛✉t♦r ❞❛ ♦❜r❛Elementos✱ ♦ q✉❛❧
❢♦✐ ♣r♦❢❡ss♦r ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❆❧❡①❛♥❞r✐❛✳
❆rq✉✐♠❡❞❡s ❞❡ ❙✐r❛❝✉s❛✱ q✉❡ é ❝♦♥s✐❞❡r❛❞♦ ♣♦r ❛❧❣✉♥s ❝♦♠♦ ♦ ♠❛✐♦r ♠❛t❡♠át✐❝♦ ❞❛ ❛♥t✐❣✉✐❞❛❞❡✱ ♠♦str♦✉ ❛ ❢ór♠✉❧❛ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❛ ár❡❛ ❞❡ ✉♠ ❝ír❝✉❧♦✱ ✉s❛♥❞♦ ♦ ▼ét♦❞♦ ❞❛ ❊①❛✉stã♦ ❡ ❛ ❝✉r✈❛ ❛t✉❛❧♠❡♥t❡ ❝❤❛♠❛❞❛ ❞❡ Espiral de Arquimedes✳
❖✉tr♦ ♣❡♥s❛❞♦r q✉❡ ❛♣❛r❡❝❡ ♥♦ ♥♦ss♦ tr❛❜❛❧❤♦ é ❇♦♥❛✈❡♥t✉r❛ ❈❛✈❛❧✐❡r✐✱ ❛✉t♦r ❞♦ ♣r✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐✱ ❛t✉♦✉ ❝♦♠♦ ♣r♦❢❡ss♦r ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇♦❧♦♥❤❛ ❡ ❞❡✐①♦✉ ♦❜r❛s ♥❛ ♠❛t❡♠át✐❝❛✱ ó♣t✐❝❛ ❡ ❛str♦♥♦♠✐❛✳
◆♦ ❝❛♣ít✉❧♦ ✷✱ ❛♣r❡s❡♥t❛♠♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ ♠❡❞✐❞❛ ❞❡ ✉♠ s❡❣♠❡♥t♦ ❡ ♠♦str❛♠♦s q✉❡ ❛ ♠❡❞✐❞❛ ❞❡ ✉♠ s❡❣♠❡♥t♦ ♣♦❞❡ s❡r ✐❣✉❛❧ ❛ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦✱ r❛❝✐♦♥❛❧ ♦✉ ✐rr❛❝✐♦♥❛❧✳ ❉❡✜♥✐♠♦s t❛♠❜é♠✱ ❞❡ ❢♦r♠❛ ❝♦♥str✉t✐✈❛✱ q✉❡ ❛ ár❡❛ ❞❡ ✉♠ r❡tâ♥❣✉❧♦ é ✐❣✉❛❧ ❛♦ ♣r♦❞✉t♦ ❞♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ r❡tâ♥❣✉❧♦ ♣❡❧❛ s✉❛ ❧❛r❣✉r❛✳ ❆♣r❡s❡♥t❛♠♦s t❛♠❜é♠ ♦ ❚❡♦r❡♠❛ ❞❡ ❇♦❧②❛✐ ✲ ●❡r✇✐❡♥✱ s❡❣✉♥❞♦ ♦ q✉❛❧ ❞♦✐s ♣♦❧í❣♦♥♦s ❞❡ ár❡❛s ✐❣✉❛✐s sã♦ ❡q✉✐❞❡❝♦♠♣♦♥í✈❡✐s✳ ❆♣r❡s❡♥t❛♠♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ ✈♦❧✉♠❡ ❡ ❞❡♠♦♥str❛♠♦s q✉❡ ♦ ✈♦❧✉♠❡ ❞❡ ✉♠ ♣❛r❛❧❡❧❡♣í♣❡❞♦ r❡tâ♥❣✉❧♦ é ✐❣✉❛❧ ❛♦ ♣r♦❞✉t♦ ❞♦ ❝♦♠♣r✐♠❡♥t♦✱ ♣❡❧❛ ❧❛r❣✉r❛ ❡ ♣❡❧❛ ❛❧t✉r❛✱ ❡ q✉❡ ✐ss♦ ✈❛❧❡ ♣❛r❛ q✉❛❧q✉❡r ♣r✐s♠❛✳
◆♦ ❝❛♣ít✉❧♦ ✸✱ ✉t✐❧✐③❛♠♦s ✉♠ ♣♦✉❝♦ s♦❜r❡ sér✐❡s ♥✉♠ér✐❝❛s ❡ s✉❛s ❛♣❧✐❝❛çõ❡s ♥❛s ❞❡♠♦♥str❛çõ❡s ❡♠ q✉❡ ♦❜t❡♠♦s ❛s ❢ór♠✉❧❛s ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞♦s ✈♦❧✉♠❡s ❞❡ ✉♠❛ ♣✐✲ râ♠✐❞❡✱ ❞❡ ✉♠ ❝♦♥❡ ❡ ❞❡ ✉♠❛ ❡s❢❡r❛✳ ❖ ♠ét♦❞♦ ✉t✐❧✐③❛❞♦ é ❝❤❛♠❛❞♦ ❞❡ ▼ét♦❞♦ ❞❡ ❊①❛✉stã♦✱ ✉s❛❞♦ ♣♦r ❆rq✉✐♠❡❞❡s✳ ❊ss❡ ♠ét♦❞♦ s❡r✈❡ ❝♦♠♦ ✉♠❛ ♦♣çã♦ ❞❡ ❞❡♠♦♥s✲ tr❛çã♦ ❞♦ ❝á❧❝✉❧♦ ❞❡ ✈♦❧✉♠❡ ❡ é ✉♠❛ ❛♣❧✐❝❛çã♦ ❞❡ ✉♠ t✐♣♦ ❞❡ sér✐❡ ❡st✉❞❛❞❛ ♣❡❧♦s ❛❧✉♥♦s ❞♦ Pr♦❢♠❛t✳
◆♦ ❝❛♣ít✉❧♦ ✹✱ ❛♣r❡s❡♥t❛♠♦s ♦ Pr✐♥❝í♣✐♦s ❞❡ ❈❛✈❛❧✐❡r✐ ❝♦♠♦ ❛①✐♦♠❛✱ ❥á q✉❡ s✉❛ ❞❡♠♦♥str❛çã♦ é ❢❡✐t❛ ❝♦♠ ❛ ✉t✐❧✐③❛çã♦ ❞❡ ❝á❧❝✉❧♦ ❛✈❛♥ç❛❞♦✱ ♦ q✉❛❧ ❢♦❣❡ à ✐❞❡✐❛ ❞❡st❡ tr❛❜❛❧❤♦ q✉❡ é ♠♦str❛r ✉♠❛ ♠❛t❡♠át✐❝❛ ♣❛r❛ s❡r ✉t✐❧✐③❛❞❛ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✳ ◆❡❧❡✱ ❞❡♠♦♥str❛♠♦s ❝♦♠♦ ♦❜t❡r ❛s ❢ór♠✉❧❛s ❞♦ ✈♦❧✉♠❡ ❞❡ ✉♠ Pr✐s♠❛✱ ❞❡ ✉♠ ❈✐❧✐♥❞r♦✱ ❞❡ ✉♠ ❈♦♥❡ ❡ ❞❡ ✉♠❛ ❊s❢❡r❛✳ ▼♦str❛♠♦s t❛♠❜é♠✱ ❛tr❛✈és ❞❡ ✉♠ ❡①❡♠♣❧♦✱ q✉❡ ♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ❛❥✉❞❛ ❛ ❡♥❝♦♥tr❛r ár❡❛s ❞❡ ✜❣✉r❛s ♣❧❛♥❛s✳
❈❛♣ít✉❧♦ ✶
❯♠ P♦✉❝♦ ❞❡ ❍✐stór✐❛
◆♦ss♦ ♣r✐♥❝✐♣❛❧ ♦❜❥❡t✐✈♦ ♥❡ss❡ ❝❛♣ít✉❧♦ é ♠♦str❛r ✉♠ ♣♦✉❝♦ ❞♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ❣❡♦♠❡tr✐❛✱ ♦ q✉❛♥t♦ ❡❧❛ ❡✈♦❧✉✐✉ ❡♠ t❡r♠♦s ❝✐❡♥tí✜❝♦s✱ ❜❡♠ ❝♦♠♦ ❢❛❧❛r ✉♠ ♣♦✉❝♦ s♦❜r❡ ❛❧❣✉♥s ❞♦s ♣r✐♥❝✐♣❛✐s ❛✉t♦r❡s q✉❡ t✐✈❡r❛♠ ♣❛rt❡ ♥❡ss❡ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦✱ q✉❡ s❡ ❞❡✉✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡✱ ♥❛ ●ré❝✐❛ ❆♥t✐❣❛✳
✶✳✶ Pr✐♠ór❞✐♦s ❞❛ ●❡♦♠❡tr✐❛
❖s ♣r✐♠❡✐r♦s ✐♥❞í❝✐♦s ❞❛ ❣❡♦♠❡tr✐❛ r✉❞✐♠❡♥t❛r ❢♦r❛♠ ♦❜s❡r✈❛❞❛s ♥♦ ❊❣✐t♦ ❛♥t✐❣♦ ❡ ♥❛ ❇❛❜✐❧ô♥✐❛✳ ❆ ❣❡♦♠❡tr✐❛ ❡r❛ ✉s❛❞❛ ❞❡ ❢♦r♠❛ ♣rát✐❝❛ ♣❛r❛ ❝❛❧❝✉❧❛r ❝♦♠♣r✐♠❡♥t♦s✱ ár❡❛s ❡ ✈♦❧✉♠❡s✳ ❈♦♠ ♦ ❛✈❛♥ç♦ ❞❛ s♦❝✐❡❞❛❞❡✱ s✉r❣✐r❛♠ ♣r♦❥❡t♦s ❞❡ ❡♥❣❡♥❤❛r✐❛ ❝♦♠♦ ❞r❡♥❛❣❡♠ ❞❡ ♣â♥t❛♥♦s✱ ❝♦♥tr♦❧❡ ❞❛s ✐♥✉♥❞❛çõ❡s ❡ ✐rr✐❣❛çõ❡s✳ ❊ss❡s ♣r♦❥❡t♦s ✜③❡r❛♠ ❝♦♠ q✉❡ ❛ ❣❡♦♠❡tr✐❛ s❡ ❞❡s❡♥✈♦❧✈❡ss❡ ❞❡ ❢♦r♠❛ ❝♦♥s✐❞❡rá✈❡❧✳ ❊r❛ ✉♠❛ ❝✐ê♥❝✐❛ ♣rát✐❝❛ q✉❡ s✉r❣✐✉ ♣❛r❛ ❛✉①✐❧✐❛r ❛s ❛t✐✈✐❞❛❞❡s ❧✐❣❛❞❛s à ❛❣r✐❝✉❧t✉r❛ ❡ à ❡♥❣❡♥❤❛r✐❛✳ ❊ss❛s ❛t✐✈✐❞❛❞❡s ♥❡❝❡ss✐t❛✈❛♠ ❞❡ ❝á❧❝✉❧♦s ♣❛r❛ ✉♠ ❝❛❧❡♥❞ár✐♦ ✉t✐❧✐③á✈❡❧✱ s✐st❡♠❛ ❞❡ ♣❡s♦s ❡ ♠❡❞✐❞❛s ♣❛r❛ ❝♦❧❤❡✐t❛✱ ♥♦ ❛r♠❛③❡♥❛♠❡♥t♦ ❡ ♥❛ ❞✐str✐❜✉✐çã♦ ❞❡ ❛❧✐♠❡♥t♦s✱ ❝r✐❛çã♦ ❞❡ ❛❣r✐♠❡♥s✉r❛ ♣❛r❛ ❝♦♥str✉✐r ❝❛♥❛✐s ❡ r❡s❡r✈❛tór✐♦s ♣❛r❛ ❞✐✈✐❞✐r ❛ t❡rr❛✳
❊ss❡s ❝á❧❝✉❧♦s ❡r❛♠ ✐♥✈❡st✐❣❛❞♦s ♣♦r s❛❝❡r❞♦t❡s q✉❡✱ ♥♦ ❊❣✐t♦ ❛♥t✐❣♦✱ ❛❞q✉✐r✐❛♠ ❝♦♥❤❡❝✐♠❡♥t♦s ❝✐❡♥tí✜❝♦s ❧✐❣❛❞♦s ❛♦ ❝❛❧❡♥❞ár✐♦ ❡ ❛♦ ❛♥♦ ❛❣rí❝♦❧❛✱ ❛tr❛✈és ❞♦s ❡st✉❞♦s ❞❛ ❆str♦♥♦♠✐❛✳
❈♦♠ ♦ ❞❡❝❧í♥✐♦ ❞❛s ❝✐✈✐❧✐③❛çõ❡s ❡❣í♣❝✐❛ ❡ ❜❛❜✐❧ô♥✐❝❛✱ ❞❡✈✐❞♦ às ♠✉❞❛♥ç❛s ♣♦❧ít✐❝❛s ❡ ❡❝♦♥ô♠✐❝❛s✱ ❝✐✈✐❧✐③❛çõ❡s ❝♦♠♦ ❛ ❣r❡❣❛ ♣❛ss❛r❛♠ ❛ s❡ ❞❡st❛❝❛r✳ ❖ ❝♦♠ér❝✐♦ ❢♦✐ ♠✉✐t♦ ✐♥❝❡♥t✐✈❛❞♦✱ ❡ ♣♦r ✈♦❧t❛ ❞♦ sé❝✉❧♦ ❱■■■ ❛✳❈✳ ♦s ❣r❡❣♦s ❞♦♠✐♥❛✈❛♠ ❛ ❛❣r✐❝✉❧t✉r❛✱ ❢✉♥❞✐❛♠ ❢❡rr♦✱ ❡s❝✉❧♣✐❛♠ ♦ ❜r♦♥③❡ ❡ ♥❛✈❡❣❛✈❛♠ ♣❡❧♦ ▼❡❞✐t❡rrâ♥❡♦✳
◆❛ ♠❛t❡♠át✐❝❛✱ q✉❡r✐❛✲s❡ ❛❧❣♦ ❛❧é♠ ❞♦s ♣r♦❝❡ss♦s ❡♠♣ír✐❝♦s✱ ♣❛r❛ ❞❡♠♦♥str❛r ❛❧❣✉♠❛s ♣r♦♣♦s✐çõ❡s✳ ❋♦✐ ♥❡ss❡ í♥t❡r✐♠ q✉❡ ♦ r❛❝✐♦♥❛❧✐s♠♦ s❡ ❞❡st❛❝♦✉✱ ❞❛♥❞♦ ê♥❢❛s❡ ❛♦ ♠ét♦❞♦ ❞❡♠♦♥str❛t✐✈♦✱ q✉❡✱ ❛té ❡♥tã♦✱ ♥ã♦ ❡r❛ ✉s❛❞♦ ♣❡❧♦s ♠❛t❡♠át✐❝♦s✱ s✉r❣✐♥❞♦ ❛ss✐♠ ❛ ♠❛t❡♠át✐❝❛ q✉❡ ❝♦♥❤❡❝❡♠♦s ❛t✉❛❧♠❡♥t❡✳
❆❧❣✉♥s ●r❛♥❞❡s P❡♥s❛❞♦r❡s ❈❛♣ít✉❧♦ ✶
❆ ✉t✐❧✐③❛çã♦ ❞♦s ♠ét♦❞♦s ❞❡♠♦♥str❛t✐✈♦s ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ♥❛s ♦❜r❛s ❞❡ ❣r❛♥✲ ❞❡s ♠❛t❡♠át✐❝♦s ❝♦♠♦ ❚❛❧❡s ❞❡ ▼✐❧❡t♦✱ P✐tá❣♦r❛s✱ ❊✉❝❧✐❞❡s ❡♥tr❡ ♦✉tr♦s✳ ❉❡st❛❝❛✲ r❡♠♦s ❛❧❣✉♥s ❞♦s ♠❛✐s ✐♠♣♦rt❛♥t❡s ♣❡♥s❛❞♦r❡s ♥♦ ❝❛♠♣♦ ❞❛ ●❡♦♠❡tr✐❛✳
✶✳✷ ❆❧❣✉♥s ●r❛♥❞❡s P❡♥s❛❞♦r❡s
❚❛❧❡s ❞❡ ▼✐❧❡t♦✱ ❝♦♥s✐❞❡r❛❞♦ ✉♠ ❞♦s s❡t❡ sá❜✐♦s ❞❛ ❆♥t✐❣✉✐❞❛❞❡✱ ❞✉r❛♥t❡ ❛ ♣r✐✲ ♠❡✐r❛ ♠❡t❛❞❡ ❞♦ sé❝✉❧♦ ❱■ ❛✳❈✳✱ ❞❡✉ ✐♥í❝✐♦ à ❣❡♦♠❡tr✐❛ ❞❡♠♦♥str❛t✐✈❛✳ ❊❧❡✱ ❛tr❛✈és ❞❡ s❡✉ ❝♦♠ér❝✐♦✱ t♦r♥♦✉✲s❡ r✐❝♦ ❡ ❞❡❞✐❝♦✉✲s❡ ❛♦s ❡st✉❞♦s ❡ ❛ ❛❧❣✉♠❛s ✈✐❛❣❡♥s✳ ❋♦✐ ❞✉r❛♥t❡ ✉♠❛ ❞❡ s✉❛s ✈✐❛❣❡♥s ♣❡❧♦ ❊❣✐t♦ q✉❡ ❚❛❧❡s ❝♦♥s❡❣✉✐✉ ❝❛❧❝✉❧❛r ❛ ❛❧t✉r❛ ❞❡ ✉♠❛ ♣✐râ♠✐❞❡ ✉s❛♥❞♦ ✉♠ ♣❡❞❛ç♦ ❞❡ ♠❛❞❡✐r❛✳ ❉❡ ✈♦❧t❛ ❛ ▼✐❧❡t♦✱ ❣❛♥❤♦✉ ❢❛♠❛✱ ❣r❛✲ ç❛s à s✉❛ ✐♥t❡❧✐❣ê♥❝✐❛ ✈❡rsát✐❧✳ ❊r❛ ✜❧ós♦❢♦✱ ❛strô♥♦♠♦ ❡ ♠❛t❡♠át✐❝♦✳ ❋♦✐ ♦ ♣r✐♠❡✐r♦ ❤♦♠❡♠ q✉❡ ❞❡♠♦♥str♦✉ q✉❡ ♦s â♥❣✉❧♦s ♦♣♦st♦s ♣❡❧♦ ✈ért✐❝❡ sã♦ ✐❣✉❛✐s✳
❋✐❣✉r❛ ✶✳✶✿ ❚❛❧❡s ❞❡ ▼✐❧❡t♦✳
❢♦♥t❡✿ ❤tt♣✿✴✴✇✇✇✳♣♦rt❛❧s❛♦❢r❛♥❝✐s❝♦✳❝♦♠✳❜r✴❛❧❢❛✴t❛❧❡s✲❞❡✲♠✐❧❡t♦✴t❛❧❡s✲❞❡✲♠✐❧❡t♦✲✶✳♣❤♣
❉❡st❛❝❛♠♦s t❛♠❜é♠ ♦ ♠❛t❡♠át✐❝♦✱ ✜❧ós♦❢♦ ❡ ♠íst✐❝♦ P✐tá❣♦r❛s q✉❡ ♥❛s❝❡✉ ♥❛ ✐❧❤❛ ❞❡ ❙❛♠♦s✱ ♣♦r ✈♦❧t❛ ❞❡ ✺✼✷ ❛✳❈✳ Pr♦✈á✈❡❧ ❞✐s❝í♣✉❧♦ ❞❡ ❚❛❧❡s✱ ❞❡✐①♦✉ ❙❛♠♦s ♣♦r ❝❛✉s❛ ❞♦ t✐r❛♥♦ P♦❧í❝r❛t❡s ❡ ❢♦✐ ♣❛r❛ ❈r♦t♦♥❛✱ ✉♠❛ ❝♦❧ô♥✐❛ ❣r❡❣❛ s✐t✉❛❞❛ ♥♦ s✉❧ ❞❛ ■tá❧✐❛✳ P✐tá❣♦r❛s ❢♦r♠♦✉ ❛ ❡s❝♦❧❛ ♣✐t❛❣ór✐❝❛✱ ❛ q✉❛❧ ❡r❛ ✉♠ ❝❡♥tr♦ ❞❡ ✜❧♦s♦✜❛✱ ♠❛t❡♠át✐❝❛ ❡ ❝✐ê♥❝✐❛s ♥❛t✉r❛✐s✳ ❊r❛ t❛♠❜é♠ ✉♠❛ s♦❝✐❡❞❛❞❡ s❡❝r❡t❛✳
➱ ❛tr✐❜✉í❞♦ ❛ P✐tá❣♦r❛s ❡ s❡✉s s❡❣✉✐❞♦r❡s ♦ ♠ér✐t♦ ❞❡ t❡r ❞❛❞♦ ♦s ♣r✐♠❡✐r♦s ♣❛ss♦s ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ❚❡♦r✐❛ ❞♦s ◆ú♠❡r♦s✱ s❡❣✉♥❞♦ ❛ q✉❛❧ ♣♦❞❡♠♦s ✐❞❡♥t✐✜❝❛r ♦s ♥ú♠❡r♦s ❛♠✐❣á✈❡✐s✱ ♦s ♥ú♠❡r♦s ✜❣✉r❛❞♦s✱ ♦s ♥ú♠❡r♦s ♣❡r❢❡✐t♦s ❡ ♦s ♥ú♠❡r♦s ❛❜✉♥❞❛♥t❡s✳ ❊♠ t❡r♠♦s ❣❡♦♠étr✐❝♦s✱ ♦s ♥ú♠❡r♦s ✜❣✉r❛❞♦s ❡①♣r❡ss❛♠ ♦ ♥ú♠❡r♦ ❞❡ ♣♦♥t♦s ❡♠ ❝❡rt❛s ❝♦♥✜❣✉r❛çõ❡s ❣❡♦♠étr✐❝❛s✱ ♦ q✉❡ ❞❡♠♦♥str❛♠ ✉♠❛ ❛ss♦❝✐❛çã♦ ❞❛ ❣❡♦♠❡tr✐❛ ❝♦♠ ❛ á❧❣❡❜r❛✳
❖s ♥ú♠❡r♦s ✜❣✉r❛❞♦s sã♦ ❞❡s❝♦❜❡rt❛s q✉❡ t❛♠❜é♠ ❢♦r❛♠ ❛tr✐❜✉í❞❛s ❛ P✐tá❣♦r❛s ❡ sã♦ r❡♣r❡s❡♥t❛❞♦s ♣❡❧♦s ♥ú♠❡r♦s q✉❛❞r❛❞♦s✱ ♥ú♠❡r♦s ♣❡♥t❛❣♦♥❛✐s ❡ ❛ss✐♠ ♣♦r
❆❧❣✉♥s ●r❛♥❞❡s P❡♥s❛❞♦r❡s ❈❛♣ít✉❧♦ ✶
❞✐❛♥t❡✳ ❈♦♠ ❡ss❡s ♥ú♠❡r♦s✱ ❢♦r❛♠ ❡st❛❜❡❧❡❝✐❞♦s ♠✉✐t♦s t❡♦r❡♠❛s ✐♥t❡r❡ss❛♥t❡s ❞❡ ♠❛♥❡✐r❛ ♣✉r❛♠❡♥t❡ ❣❡♦♠étr✐❝❛✳
❚❛♠❜é♠ ❡♥❝♦♥tr❛♠♦s ♦✉tr❛s ❝♦♥tr✐❜✉✐çõ❡s ❞❡ P✐tá❣♦r❛s ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ❣❡♦♠❡tr✐❛ ✉t✐❧✐③❛♥❞♦ ♦ tr✐â♥❣✉❧♦ ❝♦♠♦ ♦❜❥❡t♦ ❞❡ ❡st✉❞♦✳ ❆♣❡s❛r ❞♦s ❙✉♠ér✐♦s ❥á s❛❜❡r❡♠ ❛ r❡❧❛çã♦ ♠étr✐❝❛ ❡♥tr❡ ♦s ❝❛t❡t♦s ❡ ❛ ❤✐♣♦t❡♥✉s❛ ❞❡ ✉♠ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦✱ ❤á ✉♠❛ tr❛❞✐çã♦ ❡ ✉♥❛♥✐♠✐❞❛❞❡ ❡♠ ❛tr✐❜✉✐r ❛ P✐tá❣♦r❛s ❛ ❞❡s❝♦❜❡rt❛ ❞❡ss❛ r❡❧❛çã♦ q✉❡ ❧❡✈❛ ♦ s❡✉ ♥♦♠❡✿ ❚❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s ✲ q✉❡ ♦ q✉❛❞r❛❞♦ s♦❜r❡ ❛ ❤✐♣♦t❡♥✉s❛ ❞❡ ✉♠ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ é ✐❣✉❛❧ à s♦♠❛ ❞♦s q✉❛❞r❛❞♦s s♦❜r❡ ♦s ❝❛t❡t♦s✳ ❋♦✐ ✐♥❝❧✉s✐✈❡ ♥❛ ❛♣❧✐❝❛çã♦ ❞❡st❡ t❡♦r❡♠❛ q✉❡ ♦s ♣✐t❛❣ór✐❝♦s ❞❡s❝♦❜r✐r❛♠ ♦s ♥ú♠❡r♦s ✐rr❛❝✐♦♥❛✐s✱ ❡♠ q✉❡ ✈❡r✐✜❝❛r❛♠ q✉❡ ❛ ❞✐❛❣♦♥❛❧ ❞❡ ✉♠ q✉❛❞r❛❞♦ ❞❡ ❧❛❞♦ r❛❝✐♦♥❛❧ ♣♦ss✉✐ ✉♠❛ ❞✐❛❣♦♥❛❧ ♥ã♦ r❛❝✐♦♥❛❧✱ ♣♦r ❡①❡♠♣❧♦✱ ♣❛r❛ ✉♠ q✉❛❞r❛❞♦ ❞❡ ❧❛❞♦ ✶✱ t❡♠♦s ✉♠❛ ❞✐❛❣♦♥❛❧ ❝♦♠ ♠❡❞✐❞❛ ✐❣✉❛❧ ❛ √2✳
❆✐♥❞❛ ♥♦ ❊❣✐t♦✱ ❉❡♠étr✐♦✱ q✉❡ ✈✐✈❡✉ ❡♠ ❆t❡♥❛s✱ ♠✉❞❛♥❞♦ ♣❛r❛ ❆❧❡①❛♥❞r✐❛ ✐❞❡❛❧✐③♦✉ ❡ ❝r✐♦✉ ✉♠ ❝❡♥tr♦ ❞❡ s❛❜❡r ❡ ❝✉❧t✉r❛ P❛r❛ ✐ss♦✱ ♣r♦❝✉r♦✉ ❥✉♥t❛r ❣r❛♥❞❡s ♣❡♥s❛❞♦r❡s ❞♦ ♠✉♥❞♦ ❣r❡❣♦✱ ❞❡♥tr❡ ❡❧❡s ❊✉❝❧✐❞❡s✱ ♠❛t❡♠át✐❝♦ q✉❡ ♣❛ss♦✉ ❛ ❡♥s✐♥❛r ❛ ❣❡♦♠❡tr✐❛ ❡♠ ❆❧❡①❛♥❞r✐❛✳
❊✉❝❧✐❞❡s✱ ❛✉t♦r ❞❛ ❣r❛♥❞❡ ♦❜r❛ Elementos✱ ❢♦r♠❛❞❛ ♣♦r ✶✸ ❧✐✈r♦s✱ ♥❛s❝❡✉ ❡♠
❆❧❡①❛♥❞r✐❛ ❡ ❡s❝r❡✈❡✉ s❡✉ ❧✐✈r♦ ♣♦r ✈♦❧t❛ ❞❡ ✸✵✵ ❛✳❈✳ q✉❛♥❞♦ ❢♦✐ ❝❤❛♠❛❞♦ ♣❛r❛ ❢❛③❡r ♣❛rt❡ ❞♦ ♠✉s❡✉ ❞❡ ❆❧❡①❛♥❞r✐❛✱ q✉❡ ♠❛✐s t❛r❞❡ ♣❛ss♦✉ ❛ s❡r ❝❤❛♠❛❞❛ ❞❡ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❆❧❡①❛♥❞r✐❛✳ ❙✉❛ ♦❜r❛ ❡①♣õ❡ r❡s✉❧t❛❞♦s ❞❡ ✈ár✐♦s t✐♣♦s✱ ♦r❣❛♥✐③❛❞♦s s✐st❡♠❛t✐❝❛✲ ♠❡♥t❡✱ ♠✉✐t♦s ❛tr✐❜✉í❞♦s ❛ ♦✉tr♦s ❣❡ô♠❡tr❛s✳ ❆♣❡s❛r ❞✐ss♦✱ ♦s Elementos ♥ã♦ ♣♦✲
❞❡♠ s❡r ✈✐st♦s ❛♣❡♥❛s ❝♦♠♦ ✉♠❛ ❝♦♠♣✐❧❛çã♦✱ ♣♦✐s ❡ss❛ ♦❜r❛ ♣r♦♣õ❡ ✉♠ tr❛t❛♠❡♥t♦ s✐st❡♠át✐❝♦ ❡ ✉♥✐❢♦r♠❡ ❞❛ ▼❛t❡♠át✐❝❛ ❣r❡❣❛ ❜ás✐❝❛✳
❖s Elementos ❞❡ ❊✉❝❧✐❞❡s ♦❢✉s❝❛r❛♠ ♦❜r❛s ❞♦s ♠❛t❡♠át✐❝♦s ❣r❡❣♦s ❛♥t❡r✐♦r❡s
❛ ❡❧❡✱ ❞❡✈✐❞♦ ❛ s✉❛ ê♥❢❛s❡✱ ❥á q✉❡ ❝♦♥t✐♥❤❛ ✉♠❛ ❣r❛♥❞❡ ♦r❣❛♥✐③❛çã♦ ❞❡ t♦❞❛ ❛ ❣❡♦✲ ♠❡tr✐❛ q✉❡ ❤❛✈✐❛ ❛té ❡♥tã♦✳ ❉❡st❛ ❢♦r♠❛✱ ♣♦❞❡♠♦s ❞✐③❡r q✉❡ q✉❛s❡ ♥ã♦ s❡ ❝♦♥❤❡❝❡ ❢♦♥t❡s ♣r✐♠ár✐❛s ❞❛ ♠❛t❡♠át✐❝❛ ❣r❡❣❛✱ ❞✐❢❡r❡♥t❡ ❞❛ ♠❛t❡♠át✐❝❛ ♣r✐♠✐t✐✈❛ ❡❣í♣❝✐❛ ❡ ❜❛❜✐❧ô♥✐❝❛✱ q✉❡ ❡♥❝♦♥tr❛♠✲s❡ ❡♠ ✈ár✐♦s ♣❛♣✐r♦s✳
❉✐❢❡r❡♥t❡ ❞❡ ❆rq✉✐♠❡❞❡s✱ ❊✉❝❧✐❞❡s ❛❞♦t♦✉ ♦ ♠ét♦❞♦ ❛①✐♦♠át✐❝♦✲❞❡❞✉t✐✈♦✱ ❡♠ q✉❡ ❡r❛♠ ✉s❛❞♦s ❞❡✜♥✐çõ❡s✱ ❛①✐♦♠❛s ❡ ♣♦st✉❧❛❞♦s✱ q✉❡ ❡r❛♠ ❢❛t♦s ❛❝❡✐t♦s ❝♦♠♦ ❡✈✐❞❡♥t❡s ❡ ✐♥t✉✐t✐✈♦s✱ ❡ t❛♠❜é♠ ♣r♦♣♦s✐çõ❡s ✭t❡♦r❡♠❛s✮ q✉❡ ❞❡✈❡♠ s❡r ❞❡♠♦♥str❛❞❛s ❝♦♠ ❜❛s❡s ❡♠ ♣♦st✉❧❛❞♦s✱ ❛①✐♦♠❛s ❡ r❡s✉❧t❛❞♦s ❥á ❞❡♠♦♥str❛❞♦s✳
❖ r❡❧❛t♦ ❞❛s ❝♦♥tr✐❜✉✐çõ❡s ❞❡ ❆rq✉✐♠❡❞❡s ♣❛r❛ ❛ ♠❛t❡♠át✐❝❛ ♣♦❞❡♠ s❡r ❡♥❝♦♥✲ tr❛❞♦s ❡♠ ❊✈❡s ✭✷✵✵✹✮✳ ◆❛t✉r❛❧ ❞❛ ❝✐❞❛❞❡ ❣r❡❣❛ ❞❡ ❙✐r❛❝✉s❛✱ s✐t✉❛❞❛ ♥❛ ✐❧❤❛ ❞❛ ❙✐❝í❧✐❛✱ é ❝♦♥s✐❞❡r❛❞♦ ✉♠ ❞♦s ♠❛✐♦r❡s ♠❛t❡♠át✐❝♦s ❞❡ t♦❞♦s ♦s t❡♠♣♦s✳ ◆❛s❝❡✉ ♣♦r ✈♦❧t❛ ❞❡ ✷✽✼ ❛✳❈✳ ❡ ❢♦✐ ♠♦rt♦ ❡♠ ✷✶✷ ❛✳❈✳ ♣♦r ✉♠ ❝❡♥t✉r✐ã♦ r♦♠❛♥♦ ♥♦ ♠♦♠❡♥t♦ q✉❡ s❡ ❞❡❞✐❝❛✈❛ ❛ ✉♠ ❝á❧❝✉❧♦ q✉❡ ❢❛③✐❛ ♥❛ ❛r❡✐❛✳
❆rq✉✐♠❡❞❡s ❞❡✐①♦✉ ✉♠ ❧❡❣❛❞♦ q✉❡ ❢♦✐ ✉s❛❞♦ ❡ ❛♣❡r❢❡✐ç♦❛❞♦ ❛♦ ❧♦♥❣♦ ❞♦ t❡♠♣♦✱ ♣♦✐s ❝♦♥str✉çõ❡s ❝♦♠ ré❣✉❛ ❡ ❝♦♠♣❛ss♦ ♥ã♦ ♣❡r♠✐t✐❛♠ r❡s♦❧✈❡r t♦❞♦s ♦s ♣r♦❜❧❡♠❛s
❆❧❣✉♥s ●r❛♥❞❡s P❡♥s❛❞♦r❡s ❈❛♣ít✉❧♦ ✶
❋✐❣✉r❛ ✶✳✷✿ ❊stát✉❛ ❞❡ ❆rq✉✐♠❡❞❡s ❞❡ ❙✐r❛❝✉s❛✳ ❢♦♥t❡✿ ❤tt♣✿✴✴❢♦r✉♠✳t♦r✐❜❛s❤✳❝♦♠✴s❤♦✇t❤r❡❛❞✳♣❤♣❄t❂✷✵✷✶✺✸
tr❛t❛❞♦s ♣❡❧♦s ♠❛t❡♠át✐❝♦s ❣r❡❣♦s ❛♥t❡s ❡ ❞❡♣♦✐s ❞❡ ❊✉❝❧✐❞❡s✳ ❉❡st❛ ❢♦r♠❛✱ s✉r❣❡♠ ❛❧❣✉♠❛s ❝✉r✈❛s q✉❡ ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ r❡s♦❧✈❡r ♦s ♣r♦❜❧❡♠❛s ❝❧áss✐❝♦s✿ ❛ tr✐ss❡çã♦ ❞♦s â♥❣✉❧♦s✱ ❛ q✉❛❞r❛t✉r❛ ❞♦ ❝ír❝✉❧♦ ❡ ❛ ❞✉♣❧✐❝❛çã♦ ❞♦ ❝✉❜♦✱ q✉❡ ❢♦r❛♠ r❡s♦❧✈✐❞♦s s❡♠ ♦ ✉s♦ ❞❡ ré❣✉❛ ❡ ❝♦♠♣❛ss♦✳ ❈♦♥❢❡r✐r ✭ ❬✼❪✱ ✷✵✵✹❀ ❬✸❪✱ ✷✵✶✷❀ ❬✹❪✱ ✷✵✵✽✮✳
❆rq✉✐♠❡❞❡s ✉s❛✈❛ ♠ét♦❞♦s q✉❡ s❡ ❞✐❢❡r❡♥❝✐❛✈❛♠ ❞♦s ♠ét♦❞♦s ❡✉❝❧✐❞✐❛♥♦s ❡✱ ♣♦r ❡st❡ ♠♦t✐✈♦✱ ❡❧❡ ♥ã♦ ♣♦❞❡ s❡r ❝♦♥s✐❞❡r❛❞♦ ♦ s✉❝❡ss♦r ❞❡ ❊✉❝❧✐❞❡s✳ P❛r❛ ❡❧❡ ❤❛✈✐❛ ✉♠❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ♠ét♦❞♦s ❞❡ ❞❡s❝♦❜❡rt❛✱ q✉❡ ♣♦❞❡r✐❛♠ s❡r ♠❡❝â♥✐❝♦s✱ ❡ ♠ét♦❞♦s ❞❡ ❞❡♠♦♥str❛çõ❡s✱ q✉❡ ❞❡✈❡r✐❛♠ s❡r ♣✉r❛♠❡♥t❡ ❣❡♦♠étr✐❝♦s✳
❉❡♥tr❡ ❡ss❡s ♣r♦❜❧❡♠❛s ❝❧áss✐❝♦s ❡♥❢❛t✐③❛♠♦s ❛ q✉❛❞r❛t✉r❛ ❞♦ ❝ír❝✉❧♦✱ ❡♠ q✉❡ ❆rq✉✐♠❡❞❡s ❞❡♠♦♥str♦✉ q✉❡ ❛ ár❡❛ é ✐❣✉❛❧ ❛π ✈❡③❡s ❛ ♠❡❞✐❞❛ ❞♦ r❛✐♦ ❛♦ q✉❛❞r❛❞♦✱
✉s❛♥❞♦ ❛ ❈✉r✈❛ ❝❤❛♠❛❞❛ ❞❡ ❊s♣✐r❛❧ ❞❡ ❆rq✉✐♠❡❞❡s ❡ t❛♠❜é♠ ✉s❛♥❞♦ ♦ ♠ét♦❞♦ ❝❧áss✐❝♦ q✉❡ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♠ét♦❞♦ ❞❛ ❡①❛✉stã♦ ❞❡ ❊✉❞♦①♦✳
❯♠❛ ❞❡✜♥✐çã♦ ♣❛r❛ ♦ ♠ét♦❞♦ ❞❡ ❡①❛✉stã♦ ❛ q✉❛❧ ❛❞♠✐t❡ q✉❡ ✉♠❛ ❣r❛♥❞❡③❛ ♣♦ss❛ s❡r s✉❜❞✐✈✐❞✐❞❛ ✐♥❞❡✜♥✐❞❛♠❡♥t❡ ♦✉ q✉❡ é ❢♦r♠❛❞❛ ❞❡ ✉♠ ♥ú♠❡r♦ ♠✉✐t♦ ❣r❛♥❞❡ ❞❡ ♣❛rt❡s ❛tô♠✐❝❛s ✐♥❞✐✈✐sí✈❡✐s✱ ❛ ❜❛s❡ ❞♦ ♠ét♦❞♦ ❞❡ ❡①❛✉stã♦ é ❛ ♣r♦♣♦s✐çã♦✿ ✧❙❡ ❞❡ ✉♠❛ ❣r❛♥❞❡③❛ q✉❛❧q✉❡r s❡ s✉❜tr❛✐ ✉♠❛ ♣❛rt❡ ♥ã♦ ♠❡♥♦r q✉❡ s✉❛ ♠❡t❛❞❡✱ ❞♦ r❡st❛♥t❡ s✉❜tr❛✐✲s❡ t❛♠❜é♠ ✉♠❛ ♣❛rt❡ ♥ã♦ ♠❡♥♦r q✉❡ s✉❛ ♠❡t❛❞❡✱ ❡ ❛ss✐♠ ♣♦r ❞✐❛♥t❡✱ t❡r❡♠♦s✱ ♣♦r ✜♠✱ ✉♠❛ ❣r❛♥❞❡③❛ ♠❡♥♦r q✉❡ q✉❛❧q✉❡r ♦✉tr❛ ♣r❡❞❡t❡r♠✐♥❛❞❛ ❞❛ ♠❡s♠❛ ❡s♣é❝✐❡✳✧
❖✉tr♦ ♠❛t❡♠át✐❝♦ q✉❡ tr❛❜❛❧❤♦✉ ❝♦♠ ♦ ❝♦♥❝❡✐t♦ ❞❡ ✐♥❞✐✈✐s✐❜✐❧✐❞❛❞❡ ❢♦✐ ❇♦♥❛✈❡♥✲ t✉r❛ ❈❛✈❛❧✐❡r✐✱ q✉❡ ♥❛s❝❡✉ ❡♠ ▼✐❧ã♦✱ ❡♠ ✶✺✾✽✱ ❡ ❢♦✐ ❛❧✉♥♦ ❞❡ ●❛❧✐❧❡✉ ❛♦s ✶✺ ❛♥♦s✳ ❆t✉♦✉ ❝♦♠♦ ♣r♦❢❡ss♦r ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇♦❧♦♥❤❛ ❛té ❛ s✉❛ ♠♦rt❡✱ ❡♠ ✶✻✹✼✳ ❉❡✐✲ ①♦✉ ❛❧❣✉♠❛s ♦❜r❛s ♥❛ ♠❛t❡♠át✐❝❛✱ ó♣t✐❝❛ ❡ ❛str♦♥♦♠✐❛✳ P♦ré♠✱ ♦ q✉❡ r❡❛❧♠❡♥t❡ ♦ ♣r♦❥❡t♦✉ ❢♦✐ ♦ tr❛t❛❞♦Geometria Indivisibilibus ♣✉❜❧✐❝❛❞♦ ❡♠ s✉❛ ✈❡rsã♦ ✐♥✐❝✐❛❧
♥♦ ❛♥♦ ❞❡ ✶✻✸✺✳ ◆❡ss❡ tr❛❜❛❧❤♦ ❤á ✉♠❛ ❛♣r❡s❡♥t❛çã♦ ❞♦ s❡✉ ♠ét♦❞♦ ❞♦s ✐♥❞✐✈✐✲ sí✈❡✐s✱ ❝✉❥❛ ✐❞❡✐❛ r❡♠♦♥t❛ ❛ ❉❡♠ó❝r✐t♦ ❡ ❆rq✉✐♠❡❞❡s ❡ ❛ ♠♦t✐✈❛çã♦✱ t❛❧✈❡③✱ ✈❡♥❤❛ ❞❛s t❡♥t❛t✐✈❛s ❞❡ ❑❡♣❧❡r ❞❡ ❡♥❝♦♥tr❛r ❛s ár❡❛s ❡♥✈♦❧✈✐❞❛s ❡♠ s✉❛ s❡❣✉♥❞❛ ❧❡✐ ❞♦s ♠♦✈✐♠❡♥t♦s ♣❧❛♥❡tár✐♦s✱ ❡♠ q✉❡ ❡❧❡ t❡✈❡ ❞❡ r❡❝♦rr❡r ❛ ✉♠❛ ❢♦r♠❛ r✉❞✐♠❡♥t❛r ❞❡
❆❧❣✉♥s ●r❛♥❞❡s P❡♥s❛❞♦r❡s ❈❛♣ít✉❧♦ ✶
❝á❧❝✉❧♦ ✐♥t❡❣r❛❧ ✭❱✐❞❡✱ ❬✼❪✮✳
❊♠ s❡✉ tr❛t❛❞♦✱ ❈❛✈❛❧✐❡r✐ ❝♦♥❝❡✐t✉❛✈❛ ♦ t❡r♠♦ ✐♥❞✐✈✐sí✈❡❧ ❝♦♠♦✿ ✧✉♠ ✐♥❞✐✈✐sí✈❡❧ ❞❡ ✉♠❛ ♣♦rçã♦ ♣❧❛♥❛ ❞❛❞❛ é ✉♠❛ ❝♦r❞❛ ❞❡ss❛ ♣♦rçã♦ ❡ ✉♠ ✐♥❞✐✈✐sí✈❡❧ ❞❡ ✉♠ só❧✐❞♦ ❞❛❞♦ é ✉♠❛ s❡❝çã♦ ❞❡ss❡ só❧✐❞♦✧✭❬✼❪✱ ♣✳ ✹✷✺✮✳ ❉❡ss❛ ❢♦r♠❛✱ ❝♦♥s✐❞❡r❛ q✉❡ ✉♠❛ ♣♦rçã♦ ♣❧❛♥❛ ♣♦❞❡ s❡r ❢♦r♠❛❞❛ ♣♦r ✉♠❛ ✐♥✜♥✐❞❛❞❡ ❞❡ ❝♦r❞❛s ♣❛r❛❧❡❧❛s ❡ ✉♠ só❧✐❞♦ ♣♦❞❡ s❡r ❢♦r♠❛❞♦ ♣♦r ✉♠❛ ✐♥✜♥✐❞❛❞❡ ❞❡ s❡❝çõ❡s ♣❧❛♥❛s ♣❛r❛❧❡❧❛s✳ ❊ss❛s ✐❞❡✐❛s ❞❡r❛♠ ♦r✐❣❡♠ ❛♦s ❝❤❛♠❛❞♦s ♣r✐♥❝í♣✐♦s ❞❡ ❈❛✈❛❧✐❡r✐✳
❙❡❣✉♥❞♦ ❊✈❡s ✭❬✼❪✱ ♣✳ ✹✷✻✮ ✧♦s ♣r✐♥❝í♣✐♦s ❞❡ ❈❛✈❛❧✐❡r✐ r❡♣r❡s❡♥t❛♠ ❢❡rr❛♠❡♥t❛s ♣♦❞❡r♦s❛s ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❡ ár❡❛s ❡ ✈♦❧✉♠❡s ❡✱ ❛❞❡♠❛✐s✱ s✉❛ ❜❛s❡ ✐♥t✉✐t✐✈❛ ♣♦❞❡ ❢❛❝✐❧♠❡♥t❡ t♦r♥❛r✲s❡ r✐❣♦r♦s❛ ❝♦♠ ♦ ❝á❧❝✉❧♦ ✐♥t❡❣r❛❧ ♠♦❞❡r♥♦✧✳ ❉❡♣♦✐s ❞❛ ❛❝❡✐t❛çã♦ ❞❡ss❡s ♣r✐♥❝í♣✐♦s✱ ❛ ♠❛t❡♠át✐❝❛ ♣ô❞❡ r❡s♦❧✈❡r ♠✉✐t♦s ♣r♦❜❧❡♠❛s ❞❡ ♠❡♥s✉r❛çã♦ q✉❡✱ ❛té ❡♥tã♦✱ ❡r❛♠ r❡❝♦rr✐❞♦s ❛ té❝♥✐❝❛s ❛✈❛♥ç❛❞❛s ❞❡ ❝á❧❝✉❧♦ ♣❛r❛ ❛ s✉❛ r❡s♦❧✉çã♦✳
❆ ❡✈♦❧✉çã♦ ❞❛s té❝♥✐❝❛s ❞❡ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s ❢❛✈♦r❡❝❡r❛♠ ♦ ❡st✉❞♦ ❞❛ ❣❡♦♠❡tr✐❛✱ ♥♦ s❡♥t✐❞♦ ❞❡ ❝♦♥tr✐❜✉✐r ♣❛r❛ ❛ ❡✈♦❧✉çã♦ ❞♦ ❤♦♠❡♠ ♥❛ s♦❝✐❡❞❛❞❡✱ t❛♥t♦ ♥♦ ❛s♣❡❝t♦ ❡❝♦♥ô♠✐❝♦ ❝♦♠♦ ❝✉❧t✉r❛❧✳
◆❛ ❤✐stór✐❛ ❞❛ ❣❡♦♠❡tr✐❛ ❡♥❝♦♥tr❛♠♦s ❛❧❣✉♥s ❡❧❡♠❡♥t♦s q✉❡ ♥♦s ❝❤❛♠❛r❛♠ ❛ ❛t❡♥çã♦ ♣❛r❛ ✉♠ ❡st✉❞♦ ♠❛✐s ❞❡t❛❧❤❛❞♦✱ ❝♦♠♦✱ ♣♦r ❡①❡♠♣❧♦✱ ❛s ár❡❛s ❞❡ ❛❧❣✉♠❛s ✜✲ ❣✉r❛s ♣❧❛♥❛s ❡ ♦ ✈♦❧✉♠❡ ❞❡ ❛❧❣✉♥s só❧✐❞♦s ❣❡♦♠étr✐❝♦s✳ ❖ s❡✉ ❡st✉❞♦ é ❛❧❣♦ ❢❛s❝✐♥❛♥t❡ ❡✱ ♣♦r ❡st❡ ♠♦t✐✈♦✱ q✉❡ ❡s❝♦❧❤❡♠♦s ♥♦s ❛♣r♦❢✉♥❞❛r ♠❡❧❤♦r ❛❝❡r❝❛ ❞❡st❡ ❛ss✉♥t♦✳
❈❛♣ít✉❧♦ ✷
❈♦♠♣r✐♠❡♥t♦✱ ➪r❡❛ ❡ ❱♦❧✉♠❡
P♦❞❡♠♦s ♦❜s❡r✈❛r ♥♦ ❝❛♣ít✉❧♦ ❛♥t❡r✐♦r q✉❡ ❛ ❣❡♦♠❡tr✐❛ ❛❥✉❞♦✉ ♥♦ ❞❡s❡♥✈♦❧✈✐✲ ♠❡♥t♦ ❞❡ ❛❧❣✉♠❛s ❝✐✈✐❧✐③❛çõ❡s ❝♦♠♦ ❛ ❡❣í♣❝✐❛ ❡ ❜❛❜✐❧ô♥✐❝❛✱ ❡ ❝♦♠ ♦ ♣❛ss❛r ❞♦ t❡♠♣♦✱ ❡❧❛ ♣❛ss♦✉ ❛ s❡r ♠❛✐s ❡st✉❞❛❞❛ ❞❡ ❢♦r♠❛ ❛①✐♦♠át✐❝❛ ♣♦r ❛❧❣✉♥s ❣r❛♥❞❡s ♣❡♥s❛❞♦r❡s ❞❛ ❤✐stór✐❛✳
❆❧❣✉♥s ❡❧❡♠❡♥t♦s ❞❛ ❣❡♦♠❡tr✐❛✱ ❝♦♠♦ ♦ ❝♦♠♣r✐♠❡♥t♦ ❡ ❛ ár❡❛✱ ❢♦r❛♠ ♠✉✐t♦ ✉t✐❧✐③❛❞♦s ♥❛ ❢♦r♠❛çã♦ ❞❛s ♣r✐♠❡✐r❛s ❝✐✈✐❧✐③❛çõ❡s✱ ♣♦r ✐ss♦ ❞❛r❡♠♦s ê♥❢❛s❡ ❛ ❡ss❡s ❝♦♥❝❡✐t♦s✱ ♣r✐♦r✐③❛♥❞♦ s❡✉ ❛s♣❡❝t♦ ♣rát✐❝♦ ♥❛ ❝♦♥str✉çã♦ ❞❡ ♥♦✈♦s ❝♦♥❝❡✐t♦s✳ ❖✉tr❛ ❥✉st✐✜❝❛t✐✈❛ ♣❛r❛ ❡st✉❞❛r♠♦s ❡st❡s ❝♦♥❝❡✐t♦s é q✉❡ s❡r✈✐rã♦ ❞❡ ❜❛s❡ ♣❛r❛ ♦ ❡st✉❞♦ ❞❡ ✈♦❧✉♠❡s✱ q✉❡ é ✉♠ ❞♦s ❢♦❝♦s ❞❡ ♥♦ss♦ tr❛❜❛❧❤♦✳
❊ss❡s ❝♦♥❝❡✐t♦s s❡rã♦ ❞❡✜♥✐❞♦s ❞❡ ❢♦r♠❛ ❛ ❝♦♥str✉✐r ✉♠❛ s❡q✉ê♥❝✐❛ ❞✐❞át✐❝❛ ✢❡✲ ①í✈❡❧ ❝♦♠ ❛♣❧✐❝❛çõ❡s q✉❡ ♣♦❞❡rã♦ s❡r✈✐r ❞❡ ❜❛s❡ ♣❛r❛ ♣❡sq✉✐s❛❞♦r❡s ♦✉ ♣r♦❢❡ss♦r❡s ❞❡ ár❡❛s ❞❡ ✐♥t❡r❡ss❡s✳
✷✳✶ ❈♦♠♣r✐♠❡♥t♦ ❞❡ ✉♠ ❙❡❣♠❡♥t♦
◆♦ss♦ ♣r✐♠❡✐r♦ ❝♦♥❝❡✐t♦ ❛ s❡r ❡st✉❞❛❞♦ é ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ✉♠ s❡❣♠❡♥t♦✱ q✉❡ ❞❡ ❢♦r♠❛ ❤✐❡rárq✉✐❝❛✱ ✐♥tr♦❞✉③ ♦s ❞❡♠❛✐s ❝♦♥❝❡✐t♦s✳
❆♥t❡s ❞❡ ✐♥✐❝✐❛r♠♦s ❛s ❞❡✜♥✐çõ❡s é ✐♥t❡r❡ss❛♥t❡ r❡❧❡♠❜r❛r♠♦s ✉♠ ❝♦♥❝❡✐t♦ ❝r✐❛❞♦ ♥♦ sé❝✉❧♦ ❱ ❛✳❈✳ ♣❡❧♦s ♣✐t❛❣ór✐❝♦s ❛❝❡r❝❛ ❞❛ ❡①✐stê♥❝✐❛ ❞♦s ♥ú♠❡r♦s ✐rr❛❝✐♦♥❛✐s✱ ♦✉ s❡❥❛✱ ♥ú♠❡r♦s q✉❡ ♥ã♦ ♣♦❞✐❛♠ s❡r ❡s❝r✐t♦s ♥❛ ❢♦r♠❛p/q✱ ❝♦♠p❡q ✐♥t❡✐r♦s✱ ❡ q6= 0✳
❖s ♣✐t❛❣ór✐❝♦s ✈❡r✐✜❝❛r❛♠ q✉❡ ❛ ❞✐❛❣♦♥❛❧ ❞❡ ✉♠ q✉❛❞r❛❞♦ ❞❡ ❧❛❞♦ l✱ ❝♦♠ l
r❛❝✐♦♥❛❧✱ ♥ã♦ ♣♦❞❡r✐❛ s❡r ❡s❝r✐t❛ ♥❛ ❢♦r♠❛ p/q✳ ■ss♦ ❢♦✐ ♦❜s❡r✈❛❞♦ t♦♠❛♥❞♦ ✉♠
q✉❛❞r❛❞♦ ❞❡ ❧❛❞♦ r❛❝✐♦♥❛❧p/q✱ ❡♥tã♦✱ ♣ô❞❡✲s❡✱ ♣♦r r❡❞✉çã♦ ❛♦ ❛❜s✉r❞♦✱ ♠♦str❛r q✉❡
s✉❛ ❞✐❛❣♦♥❛❧ ♥ã♦ ♣♦❞✐❛ s❡r ❡s❝r✐t❛ ♥❛ ❢♦r♠❛ p/q✳ ❱❡❥❛♠♦s ❛ ❞❡♠♦♥str❛çã♦✳
❙✉♣♦♥❤❛ q✉❡ ❛ ❞✐❛❣♦♥❛❧ ❞♦ q✉❛❞r❛❞♦ ❞❡ ❧❛❞♦p/q s❡❥❛ ✉♠ ♥ú♠❡r♦ r❛❝✐♦♥❛❧ ❡s❝r✐t♦
♥❛ ❢♦r♠❛ m/n✱ s❡♥❞♦ m ❡ n ♥ú♠❡r♦s ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s✳ ❉❛í✱ ✉s❛♥❞♦ ♦ t❡♦r❡♠❛ ❞❡
P✐tá❣♦r❛s✱ t❡r❡♠♦s (m/n)2 = (p/q)2+ (p/q)2✱ ♦✉ s❡❥❛✱ (m/n)2 = 2·(p/q)2 ❡♠ q✉❡
♦❜t❡♠♦s(qm)2
= 2·(np)2✳ ❚♦❞❛✈✐❛✱ ♦❜s❡r✈❡ q✉❡ ♥♦ ♣r✐♠❡✐r♦ ♠❡♠❜r♦ só ♣♦❞❡♠♦s
❈♦♠♣r✐♠❡♥t♦ ❞❡ ✉♠ ❙❡❣♠❡♥t♦ ❈❛♣ít✉❧♦ ✷
t❡r ✉♠❛ q✉❛♥t✐❞❛❞❡ ♣❛r ❞❡ ♣♦tê♥❝✐❛s ❞❡ ✷ ❡ ♥♦ s❡❣✉♥❞♦ ♠❡♠❜r♦ ✉♠❛ q✉❛♥t✐❞❛❞❡ í♠♣❛r ❞❡ ♣♦tê♥❝✐❛s ❞❡ ✷ ♦ q✉❡ é ✉♠ ❛❜s✉r❞♦✳ P♦rt❛♥t♦✱ ❛ ❞✐❛❣♦♥❛❧ ❞♦ q✉❛❞r❛❞♦ ♥ã♦ ♣♦❞❡ s❡r ❡s❝r✐t❛ ❝♦♠♦p/q✳
❆ ❞❡s❝♦❜❡rt❛ ❞♦s ♥ú♠❡r♦s ✐rr❛❝✐♦♥❛✐s só ❢♦✐ ♣♦ssí✈❡❧ ❣r❛ç❛s ❛♦s ♣✐t❛❣ór✐❝♦s q✉❡ t✐♥❤❛♠ ❝♦♥❤❡❝✐♠❡♥t♦ ❞♦ ❝♦♥❝❡✐t♦ ❞❡ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ✉♠ s❡❣♠❡♥t♦ ❡ q✉❡✱ ❞❡ ❢♦r♠❛ r✐❣♦r♦s❛✱ r❡q✉❡r ✉♠ ❝✉✐❞❛❞♦ ♠❛✐s ❡s♣❡❝í✜❝♦ ♣❛r❛ s❡r r❡♣r❡s❡♥t❛❞♦✳
❱❛♠♦s r❡♣r❡s❡♥t❛r ✉♠ s❡❣♠❡♥t♦ ❢♦r♠❛❞♦ ♣♦r ❞♦✐s ♣♦♥t♦s✱A ❡B✱ ♣♦r AB ❡ s✉❛
♠❡❞✐❞❛✱ ♦✉ ❝♦♠♣r✐♠❡♥t♦✱ ♣♦r AB✳ ❆ ♠❡❞✐❞❛ ❞♦ s❡❣♠❡♥t♦ AB é ✉♠ ♥ú♠❡r♦ q✉❡
❞❡✈❡ ❡①♣r❡ss❛r q✉❛♥t❛s ✈❡③❡s ♦ s❡❣♠❡♥t♦ AB ❝♦♥té♠ ✉♠ s❡❣♠❡♥t♦u✱ ♦ q✉❛❧ ♣♦ss✉✐
♠❡❞✐❞❛ ✐❣✉❛❧ ❛ 1❡✱ ♣♦r ❡st❡ ♠♦t✐✈♦✱ ❝❤❛♠❛r❡♠♦s ❞❡ s❡❣♠❡♥t♦ ✉♥✐tár✐♦✳
❊①❡♠♣❧♦✿ ❙❡ ❢♦r❡♠ ❞❛❞♦s três ♣♦♥t♦s ❝♦❧✐♥❡❛r❡s A✱ B ❡ C✱ ♦♥❞❡ AB✱ BC sã♦
❝♦♥❣r✉❡♥t❡s ❛u✱ ❡♥tã♦ AC =AB+BC = 2✳
❋✐❣✉r❛ ✷✳✶✿ ❙❡❣♠❡♥t♦ ❢♦r♠❛❞♦ ♣❡❧♦s ♣♦♥t♦s A✱ B ❡C✳
❊♠ ❣❡r❛❧✱ s❡ ❢♦r ♣♦ssí✈❡❧ ♦❜t❡r n + 1 ♣♦♥t♦s ❝♦❧✐♥❡❛r❡s A1, A2, ..., An, An+1✱ ❞❡
s❡❣♠❡♥t♦s ❝♦♥❣r✉❡♥t❡s ❛♦ s❡❣♠❡♥t♦ ✉♥✐tár✐♦u✱ ❡♥tã♦ ❛ ♠❡❞✐❞❛ ❞❡A1An+1 s❡rá ✐❣✉❛❧
❛n✳ ◆❡st❡ ❝❛s♦ ❡s❝r❡✈❡♠♦s✳
A1A2+A2A3+· · ·+AnAn+1 =n.
❚♦♠❛♥❞♦A1 =A ❡An+1 =B✱ t❡r❡♠♦s ❛ss✐♠AB =n✱ ❝♦♠♦ ✐❧✉str❛❞♦ ♥❛ ❋✐❣✉r❛
✷✳✷ ❛ s❡❣✉✐r✳
❋✐❣✉r❛ ✷✳✷✿ ❙❡❣♠❡♥t♦ AB ❢♦r♠❛❞♦ ♣♦rn s❡❣♠❡♥t♦s ❝♦♥❣r✉❡♥t❡s ❛ u✳
❆ ♠❡❞✐❞❛ ❞♦ s❡❣♠❡♥t♦ AB r❡♣r❡s❡♥t❛ q✉❛♥t❛s ✈❡③❡s u ❝❛❜❡ ♥♦ s❡❣♠❡♥t♦ AB✳
■st♦ ✈❛❧❡ q✉❛♥❞♦ ✉♠ s❡❣♠❡♥t♦AB é ♠ú❧t✐♣❧♦ ❞❡ u✳ P♦ré♠✱ s❡ ♦ s❡❣♠❡♥t♦ AB ❢♦r
♠❡♥♦r q✉❡ ♦ s❡❣♠❡♥t♦ ✉♥✐tár✐♦❄ Pr✐♠❡✐r♦✱ s✉♣♦♥❤❛ q✉❡ ❡①✐st❛ ✉♠ s❡❣♠❡♥t♦w✱ t❛❧
q✉❡ ❡st❡❥❛n ✈❡③❡s ❝♦♥t✐❞❛ ♥♦ s❡❣♠❡♥t♦ u❡ ❡st❡❥❛m✈❡③❡s ❝♦♥t✐❞❛ ♥♦ s❡❣♠❡♥t♦ AB✱
s❡♥❞♦ m ❡ n ♥ú♠❡r♦s ✐♥t❡✐r♦s✳ ❉✐③❡♠♦s q✉❡ w é s✉❜♠ú❧t✐♣❧♦ ❝♦♠✉♠ ❞❡ AB ❡ u✱ ❡
♣♦r ❡st❡ ♠♦t✐✈♦✱AB ❡ u sã♦ ❝❤❛♠❛❞♦s ❞❡ ❝♦♠❡♥s✉rá✈❡✐s✱ ❡ AB =m/n✱ ❥á q✉❡ w é
1/n ♣❛rt❡ ❞❡ u✱ ❡AB é m ✈❡③❡s ❛ ♣❛rt❡ ❞❡ w✱ ♦✉ s❡❥❛✱ m ✈❡③❡s 1/n✳
➪r❡❛ ❞❡ ✉♠ ❘❡tâ♥❣✉❧♦ ❈❛♣ít✉❧♦ ✷
❆❣♦r❛✱ s❡ ♦ s❡❣♠❡♥t♦ é ✐♥❝♦♠❡♥s✉rá✈❡❧ ❝♦♠ ♦ s❡❣♠❡♥t♦ ✉♥✐tár✐♦ u✱ ♦✉ s❡❥❛✱
♥ã♦ é ♣♦ssí✈❡❧ ❡s❝r❡✈❡r s✉❛ ♠❡❞✐❞❛ ♥❛ ❢♦r♠❛m/n✱ ❡♥tã♦ s✉❛ ♠❡❞✐❞❛ é ✉♠ ♥ú♠❡r♦
✐rr❛❝✐♦♥❛❧✳ ▼❛s✱ ♦ q✉❡ é ✉♠ ♥ú♠❡r♦ ✐rr❛❝✐♦♥❛❧❄ ❙❡❣✉♥❞♦ ▲✐♠❛ ✭❬✶✵❪✱ ✷✵✵✾✱ ♣✳ ✹✮ ✧❆ r❡s♣♦st❛ ♥ã♦ é ♠✉✐t♦ s✐♠♣❧❡s✳ ❊♥q✉❛♥t♦ ✉♠ ♥ú♠❡r♦ r❛❝✐♦♥❛❧ t❡♠ ✉♠❛ ❡①♣r❡ssã♦ ✧❡①❛t❛✧ ❝♦♠♦ ♦ q✉♦❝✐❡♥t❡ p/q ❞❡ ❞♦✐s ♥ú♠❡r♦s ✐♥t❡✐r♦s✱ ✉♠ ♥ú♠❡r♦ ✐rr❛❝✐♦♥❛❧ ✜❝❛
❞❡t❡r♠✐♥❛❞♦ q✉❛♥❞♦ s❡ ❝♦♥❤❡❝❡♠ s❡✉s ✈❛❧♦r❡s ❛♣r♦①✐♠❛❞♦s✳✧
❚♦♠❡♠♦s ✉♠ s❡❣♠❡♥t♦ AB q✉❡ s❡❥❛ ✐♥❝♦♠❡♥s✉rá✈❡❧ ❝♦♠ ❛ ✉♥✐❞❛❞❡ ❞❡ ❝♦♠♣r✐✲
♠❡♥t♦ u✱ ❧♦❣♦ ❛ ♠❡❞✐❞❛ ❞❡ AB é ✉♠ ♥ú♠❡r♦ ✐rr❛❝✐♦♥❛❧✳
❙❡❥❛ ❞❛❞♦n ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦✳ ❉✐✈✐❞✐♥❞♦ ♦ s❡❣♠❡♥t♦ ✉♥✐tár✐♦ u ❡♠ n
♣❛rt❡s ✐❣✉❛✐s✱ q✉❡ ❝❤❛♠❛r❡♠♦s ❞❡ w ✉♠❛ ❞❡ss❛s ♣❛rt❡s✳ ❊①✐st❡ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦
♣♦s✐t✐✈♦m✱ ❡♠ q✉❡AB ♣♦ss✉✐ ✉♠❛ q✉❛♥t✐❞❛❞❡m❞❡ s❡❣♠❡♥t♦s ❝♦♥❣r✉❡♥t❡s ❡ ❛✐♥❞❛
❢❛❧t❛ ❛❧❣♦ ♣❛r❛ ❝♦♠♣❧❡t❛r✱ ♣♦ré♠✱ m + 1 q✉❛♥t✐❞❛❞❡s ❞❡ s❡❣♠❡♥t♦s ✐❣✉❛✐s ❛ 1/n
❢♦r♠❛ ✉♠ s❡❣♠❡♥t♦ ♠❛✐♦r q✉❡ ♦ s❡❣♠❡♥t♦AB✳ ❉❡st❛ ❢♦r♠❛✱ ♦❜t❡♠♦s✿
m
n < AB <
m+ 1
n .
▲♦❣♦✱ ♦ ♥ú♠❡r♦ m/n é ✉♠❛ ❛♣r♦①✐♠❛çã♦ ♣♦r ❢❛❧t❛ ♣❛r❛ ❛ ♠❡❞✐❞❛ ❞♦ s❡❣♠❡♥t♦ AB✱ ❝♦♠ ❡rr♦ ✐♥❢❡r✐♦r ❛ 1/n✳ ❉❛ ♠❡s♠❛ ❢♦r♠❛✱ (m+ 1)/n é ✉♠❛ ❛♣r♦①✐♠❛çã♦ ♣♦r
❡①❝❡ss♦ ❞♦ ♥ú♠❡r♦ ✐rr❛❝✐♦♥❛❧AB✱ ❝♦♠ ❡rr♦ ✐♥❢❡r✐♦r ❛ 1/n✳
✷✳✷ ➪r❡❛ ❞❡ ✉♠ ❘❡tâ♥❣✉❧♦
❆ ♣❛rt✐r ❞♦s ❝♦♥❝❡✐t♦s ❞❡ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ✉♠ s❡❣♠❡♥t♦✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ♦s ❡❧❡♠❡♥t♦s q✉❡ ❝♦♠♣õ❡ ❛s ❞✐♠❡♥sõ❡s ❞❡ ✉♠ r❡tâ♥❣✉❧♦✳ ❖ ❝♦♥❝❡✐t♦ ❞❡ ár❡❛ ✈❡♠ ❞❛ ❝♦♠♣❛r❛çã♦ ❡♥tr❡ ✜❣✉r❛s ♣❧❛♥❛s✱ ❡♠ q✉❡ ❛ ár❡❛ ❞❡ q✉❛❧q✉❡r ✜❣✉r❛ é ❞❛❞❛ ❡♠ ❢✉♥çã♦ ❞❡ ✉♠ q✉❛❞r❛❞♦ ❞❡ ❧❛❞♦1✳ P♦r ❡①❡♠♣❧♦✱ ❛ ár❡❛ ❞❡ ✉♠ r❡tâ♥❣✉❧♦ ❞❡ ❧❛❞♦s3m ❡4m
❝❛❜❡♠ ✶✷ q✉❛❞r❛❞♦s ❞❡ ár❡❛ ✐❣✉❛❧ ❛ 1m2✱ ❝♦♠♦ ♣♦❞❡♠♦s ✈❡r ♥❛ ❋✐❣✉r❛ ✷✳✸✳
❋✐❣✉r❛ ✷✳✸✿ ❘❡tâ♥❣✉❧♦ ❞❡ ❧❛❞♦s 3m ❡ 4m✳
❖s ❝♦♥❝❡✐t♦s ✈✐st♦s ♥❡ss❡ ❡①❡♠♣❧♦ s❡r✈✐rã♦ ❞❡ ❜❛s❡ ♣❛r❛ ❛ ❞❡♠♦♥str❛çã♦ ❞❛ s❡✲ ❣✉✐♥t❡ ♣r♦♣♦s✐çã♦✳
➪r❡❛ ❞❡ ✉♠ ❘❡tâ♥❣✉❧♦ ❈❛♣ít✉❧♦ ✷
Pr♦♣♦s✐çã♦ ✷✳✷✳✶ ❆ ár❡❛ ❞❡ ✉♠ r❡tâ♥❣✉❧♦ é ✐❣✉❛❧ ❛♦ ♣r♦❞✉t♦ ❞❛ ❜❛s❡ ♣❡❧❛ ❛❧t✉r❛ ❞♦ r❡tâ♥❣✉❧♦✳
Pr♦✈❛✿ ❈♦♥s✐❞❡r❡ ✉♠ q✉❛❞r❛❞♦ ✉♥✐tár✐♦✱ ♦✉ s❡❥❛✱ ❞❡ ❧❛❞♦1✱ q✉❡ ❞✐✈✐❞✐r❡♠♦s ❡♠N
♣❛rt❡s ✐❣✉❛✐s✱ ❡♠ q✉❡N é ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧✳ ❙❡❥❛q❛ ♠❡❞✐❞❛ ❞♦ ❧❛❞♦ ❞♦ q✉❛❞r❛❞♦
♠❡♥♦r ❢♦r♠❛❞♦ ♣❡❧❛ ❞✐✈✐sã♦ ❞♦ q✉❛❞r❛❞♦ ✉♥✐tár✐♦ ♣♦r N✱ ♦✉ s❡❥❛✱ q = 1
N✱ s❡♥❞♦
❛ss✐♠✱q2
= 1
N2✳
❋✐❣✉r❛ ✷✳✹✿ ◗✉❛❞r❛❞♦ ❞❡ ❧❛❞♦ 1❞✐✈✐❞✐❞♦ ❡♠ N ♣❛rt❡s✳
❉❛❞♦s x ❡ y ❛s ♠❡❞✐❞❛s ❞♦s ❧❛❞♦s ❞❡ ✉♠ r❡tâ♥❣✉❧♦ ❝♦♠♦ ♣♦❞❡♠♦s ♦❜s❡r✈❛r ♥❛
❋✐❣✉r❛ ✷✳✺✳ ❊♥tã♦ ♣♦❞❡♠♦s ✐♥❞✐❝❛r ♣♦r m ♦ ♥ú♠❡r♦ ✐♥t❡✐r♦ ❞❛ ❞✐✈✐sã♦ ❞❡ x ♣♦r q
❡ n ♦ ♥ú♠❡r♦ ✐♥t❡✐r♦ ❞❛ ❞✐✈✐sã♦ ❞❡ y ♣♦r q✱ ❡ ♦ ♥ú♠❡r♦ ❞❡ q✉❛❞r❛❞♦s ❝♦♥t✐❞♦s ♥♦
r❡tâ♥❣✉❧♦ s❡rá✱ ♥♦ ♠í♥✐♠♦✱m·n✱ ❡♥q✉❛♥t♦ q✉❡ ♦ ♥ú♠❡r♦ ❞❡ q✉❛❞r❛❞♦s q✉❡ ❝♦♥té♠
♦ r❡tâ♥❣✉❧♦ ♥ã♦ s❡rá ♠❛✐♦r q✉❡(m+ 1)(n+ 1)✳ ❉❛í r❡s✉❧t❛ q✉❡ ❛ ár❡❛ ❞♦ r❡tâ♥❣✉❧♦
❡stá ❝♦♠♣r❡❡♥❞✐❞❛ ❡♥tr❡m·n·q2 ❡
(m+ 1)·(n+ 1)·q2✱ ♦✉ s❡❥❛✱
❋✐❣✉r❛ ✷✳✺✿ ❘❡tâ♥❣✉❧♦ ❞❡ ❧❛❞♦s ✐rr❛❝✐♦♥❛✐s✳
m·n·q2 ≤❆ <(mn+m+n+ 1)·q2.
❉❡♠♦♥str❛r❡♠♦s ❛❣♦r❛ q✉❡ ♦ ♣r♦❞✉t♦ ❞❡x·y❡stá ❝♦♠♣r❡❡♥❞✐❞♦ ❡♥tr❡ ♦s ♠❡s♠♦s
♥ú♠❡r♦s✳ ❊❢❡t✐✈❛♠❡♥t❡✱ t❡♠✲s❡ m·q ≤x <(m+ 1)·q ❡n·q≤y <(n+ 1)·q✳
P♦r ❡st❡ ♠♦t✐✈♦✱ t❡♠♦s✿
m·n·q2
≤xy <(m+ 1)·(n+ 1)·q2
,
