POLIEDROS REGULARES DO ENSINO MÉDIO

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(1)❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛ ❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ◆❛t✉r❡③❛ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚ P♦❧✐❡❞r♦s r❡▼❣✉é❧❞❛✐r♦❡†s ♥♦ ❊♥s✐♥♦ ♣♦r ❍❡r❝✉❧❡s ❞♦ ◆❛s❝✐♠❡♥t♦ ❙✐❧✈❛ s♦❜ ♦r✐❡♥t❛çã♦ ❞❛ Pr♦❢➟ ❉r➟ ▼✐r✐❛♠ ❞❛ ❙✐❧✈❛ P❡r❡✐r❛ ❚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✐❝❛✱ ❥✉♥t♦ ❛♦ Pr♦✲ ❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ Pr♦✜ss✐♦♥❛❧ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ✲ P❘❖❋✲ ▼❆❚✱ ❞♦ ❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ◆❛t✉r❡③❛✱ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛✲ r❛í❜❛✳ ❆❣♦st♦✴✷✵✶✹ ❏♦ã♦ P❡ss♦❛ ✲ P❇ †❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❢♦✐ r❡❛❧✐③❛❞♦ ❝♦♠ ❛♣♦✐♦ ❞❛ ❈❆P❊❙✱ ❈♦♦r❞❡♥❛çã♦ ❞❡ ❆♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❡ P❡ss♦❛❧ ❞❡ ◆í✈❡❧ ❙✉♣❡r✐♦r✳

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(4) ❆❣r❛❞❡❝✐♠❡♥t♦s ❖s ❛❣r❛❞❡❝✐♠❡♥t♦s sã♦ ✐♥ú♠❡r♦s✱ ♣♦✐s ❢♦r❛♠ t❛♥t❛s ♣❡ss♦❛s q✉❡ ♠❡ ✜③❡r❛♠ ❝❤❡✲ ❣❛r ❛ ❡ss❡ ♠♦♠❡♥t♦✳ Pr✐♠❡✐r❛♠❡♥t❡ ❛ ❉❡✉s ❞✉r❛♥t❡ ❡ss❡s ❞♦✐s ❛♥♦s ❡ ♠❡✐♦ s❡♠♣r❡ ❡st❡✈❡ ♣r❡s❡♥t❡ ❝♦♠♦ ❢♦♥t❡ ❞❡ ✐♥s♣✐r❛çã♦ ❡ ♣r♦t❡çã♦ ♥❛s t❛♥t❛s ❡ t❛♥t❛s ✈✐❛❣❡♥s✱ ♣r♦✈❛s✳ ❙❡✐ q✉❡ ❡♠ t♦❞♦s ♦s ♠♦♠❡♥t♦s ❊❧❡ ❡st❛✈❛✱ ❡stá ❡ ❡st❛rá ❣✉✐❛♥❞♦✲♠❡✳ ❆♦s ♠❡✉s ❢❛♠✐❧✐❛r❡s q✉❡ s❡♠♣r❡ ♠❡ ❛❥✉❞❛r❛♠ ♥♦s ♠♦♠❡♥t♦s ♠❛✐s ❞✐❢í❝❡✐s ❞❡ss❛ ❥♦r♥❛❞❛✱ s❡❥❛ ❝♦♠ ✉♠❛ ♣❛❧❛✈r❛ ❞❡ ✐♥❝❡♥t✐✈♦✱ ❝♦♠ ♦r❛çõ❡s ♦✉ ✈✐♥❞♦ ❝♦♠✐❣♦ ❡♠ ❞✐❛s ❞❡ ♣r♦✈❛✱ ❡♠ ❡s♣❡❝✐❛❧ ❛ ♠✐♥❤❛ t✐❛ ▼❛r✐❛✱ ♠❡✉s ✐r♠ã♦s ❏✉❝❡❧♦ ❡ ■♦♥❛r❛ ❡ ❛♦s ♠❡✉s ♣❛✐s ❏♦sé P❡❞r♦ ❡ ❍♦s❛♥❛ q✉❡ s❡♠♣r❡ ♠❡ ❛♣♦✐❛r❛♠ ♥♦s ❡st✉❞♦s ❡ ♠❡ ❡♥s✐♥❛r❛♠ q✉❡ ♣❛r❛ ❝♦♥q✉✐st❛r ❛❧❣♦ ♥❡ss❛ ✈✐❞❛ é ♣r❡❝✐s♦❀ ♣❛❝✐ê♥❝✐❛✱ ❤♦♥❡st✐❞❛❞❡ ❡ ❞❡❞✐❝❛çã♦✳ ❆ ♠✐♥❤❛ ❡s♣♦s❛ ❈r✐st✐♥❛ ❝❡rt❛♠❡♥t❡ ❛ ♣❡ss♦❛ q✉❡ ♠❛✐s ♠❡ ✐♥❝❡♥t✐✈♦✉ ♥❡ss❡ ♣❡✲ rí♦❞♦✱ ♠❡s♠♦ ♥♦s ♠♦♠❡♥t♦s q✉❡ ♣❡♥s❡✐ ❡♠ ❞❡s✐st✐r s✐♠♣❧❡s♠❡♥t❡ ✈♦❝ê ❢❛❧❛✈❛ ❛❧❣♦ ❡ ✐ss♦ ❜❛st❛✈❛ ♣r❛ ♠❡ tr❛③❡r â♥✐♠♦ ❡ ❝♦♥t✐♥✉❛r✳ ❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ❝♦❧❡❣❛s ❞❡ t✉r♠❛ ❛♦s q✉❡ ❝❤❡❣❛r❛♠ ❛té ❡ss❡ ♠♦♠❡♥t♦ ♦✉ q✉❡ ♣♦r ❛❧❣✉♠ ♠♦t✐✈♦ ❞❡s✐st✐r❛♠ ♥♦ ❝❛♠✐♥❤♦✱ ♥❛s ❜r✐♥❝❛❞❡✐r❛s✱ ♥♦s ❜❛t❡ ♣❛♣♦s✱ ♥♦ ❡st✉❞♦✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❛♦s ❛♠✐❣♦s ❙❛❧❛t✐❡❧✱ ❋é❧✐①✱ ❉♦✈❛❧ ❡ ◆❡t♦ s❛✐❜❛♠ q✉❡ ✈♦❝ês ❢♦r❛♠ ❡ss❡♥❝✐❛✐s ♣❛r❛ ❡ss❛ ❝♦♥q✉✐st❛✳ ❆♦s ❝♦❧❡❣❛s ❞❡ tr❛❜❛❧❤♦ ❡ ❛❧✉♥♦s ❞❛ ❊❘❊▼ ❇❡♥❡❞✐t❛ ❞❡ ▼♦r❛✐s ●✉❡rr❛ q✉❡ ❝♦♠ ♣❛❧❛✈r❛s ❞❡ ✐♥❝❡♥t✐✈♦✱ ♠❡♥s❛❣❡♥s✱ ❧✐❣❛çõ❡s ❡ ♦r❛çõ❡s ❢♦r❛♠ ❛❧✐❝❡r❝❡s ♣❛r❛ ♠❡ ♠❛♥t❡r ✜r♠❡ ❡♠ ♠♦♠❡♥t♦s ❞❡ ❢r❛q✉❡③❛✳ ➚ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s ❞♦ ♠❡str❛❞♦ q✉❡ ❝♦♠ ❞❡❞✐❝❛çã♦ ❡ ♣❛❝✐ê♥❝✐❛ ♥♦s ♣r♦♣♦r✲ ❝✐♦♥❛r❛♠ ♦ ❝❛♠✐♥❤♦ ❛ ♥♦✈♦s ❝♦♥❤❡❝✐♠❡♥t♦s✱ ❡♠ ❡s♣❡❝✐❛❧ ❛ ♠✐♥❤❛ ♦r✐❡♥t❛❞♦r❛ ❛ ♣r♦❢❡ss♦r❛ ▼✐r✐❛♠✱ q✉❡ ❛❧é♠ ❞❡ ♣❛❝✐❡♥t❡ ❡ ❞❡❞✐❝❛❞❛✱ s✉❛s ♦r✐❡♥t❛çõ❡s ❡ ❝♦♥s❡❧❤♦s ❢♦r❛♠ ♠❛✐s q✉❡ ❢✉♥❞❛♠❡♥t❛✐s ❡ ♦ ♣r♦❢❡ss♦r ❋❧❛♥❦ q✉❡ t❛♠❜é♠ ❢♦✐ ❞❡ ❢✉♥❞❛♠❡♥t❛❧ ✐♠♣♦rtâ♥❝✐❛ ♣❛r❛ ❛ ♦r❣❛♥✐③❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳ ➚ t♦❞♦s q✉❡ ❞✐r❡t❛ ❡ ✐♥❞✐r❡t❛♠❡♥t❡ ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ ❛ ❝♦♥❝❧✉sã♦ ❞❡ss❛ ❡t❛♣❛ ❞❡ ♠✐♥❤❛ ✈✐❞❛✳ ✐✈

(5) ❉❡❞✐❝❛tór✐❛ ➚ t♦❞♦s q✉❡ ❞❡❞✐❝❛♠✲s❡ ♣❛r❛ ✉♠❛ ❡❞✉❝❛çã♦ ❞❡ q✉❛❧✐❞❛❞❡✳ ✳ ✈

(6) ❘❡s✉♠♦ ◆❡st❡ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛♠♦s ✉♠ ❡st✉❞♦ s♦❜r❡ ♦s ♣♦❧✐❡❞r♦s r❡❣✉❧❛r❡s✱ ❝♦♠♣❛r❛♥❞♦ ❡ ❞✐s❝✉t✐♥❞♦ ♦s ❝♦♥❝❡✐t♦s ❡ ❛s ❞❡✜♥✐çõ❡s q✉❡ sã♦ ❞❛❞❛s ♥♦ ❡st✉❞♦ ❞♦s ♣♦❧✐❡❞r♦s r❡✲ ❣✉❧❛r❡s ♥♦s ❧✐✈r♦s ❞✐❞át✐❝♦s ♠❛✐s ✉t✐❧✐③❛❞♦s ♥❛s ❡s❝♦❧❛s ❜r❛s✐❧❡✐r❛s ❞❡ ❊♥s✐♥♦ ▼é❞✐♦✳ Pr♦✈❛♠♦s ♦ t❡♦r❡♠❛ ❞❡ ❊✉❧❡r✱ ❝❛❧❝✉❧❛♠♦s ár❡❛s ❞❡ s✉♣❡r❢í❝✐❡s ❡ ♦s ✈♦❧✉♠❡s ❞♦s ♣♦❧✐✲ ❡❞r♦s r❡❣✉❧❛r❡s✳ P♦r ✜♠✱ ❛♣r❡s❡♥t❛♠♦s ❛❧❣✉♥s s♦❢t✇❛r❡s ♠❛t❡♠át✐❝♦s q✉❡ ♣♦❞❡♠ s❡r ✉t✐❧✐③❛❞♦s ♣❡❧♦s ❛❧✉♥♦s ❡ ♣r♦❢❡ss♦r❡s ❞❡ ▼❛t❡♠át✐❝❛ ♥❛s ❛✉❧❛s ❞❡ ❣❡♦♠❡tr✐❛ ❡s♣❛❝✐❛❧ ❝♦♠♦ ♠❛t❡r✐❛❧ ❛✉①✐❧✐❛r ♥♦ ♣r♦❝❡ss♦ ❞❡ ❡♥s✐♥♦ ❡ ❛♣r❡♥❞✐③❛❣❡♠ ❞❡st❡ t❡♠❛ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛✳ P❛❧❛✈r❛s✲❝❤❛✈❡s✿ ●❡♦♠❡tr✐❛ ❡s♣❛❝✐❛❧✱ P♦❧✐❡❞r♦s r❡❣✉❧❛r❡s✱ ❚❡♦r❡♠❛ ❞❡ ❊✉❧❡r✱ ❙♦❢t✇❛r❡s ❞❡ ▼❛t❡♠át✐❝❛✳ ✈✐

(7) ❆❜str❛❝t ■♥ t❤✐s ✇♦r❦ ✇❡ ♣r❡s❡♥t ❛ st✉❞② ♦❢ t❤❡ r❡❣✉❧❛r ♣♦❧②❤❡❞r❛✱ ❝♦♠♣❛r✐♥❣ ❛♥❞ ❞✐s❝✉s✲ s✐♥❣ t❤❡ ❝♦♥❝❡♣ts ❛♥❞ ❞❡✜♥✐t✐♦♥s ❣✐✈❡♥ ✐♥ t❤❡ st✉❞② ♦❢ r❡❣✉❧❛r ♣♦❧②❤❡❞r❛ ✐♥ t❡①t❜♦♦❦s ♠♦st ✇✐❞❡❧② ✉s❡❞ ✐♥ ❇r❛③✐❧✐❛♥ ❤✐❣❤ s❝❤♦♦❧s✳ ❲❡ ♣r♦✈❡ t❤❡ t❤❡♦r❡♠ ♦❢ ❊✉❧❡r✱ ✇❡ ❝❛❧✲ ❝✉❧❛t❡ s✉r❢❛❝❡ ❛r❡❛s ❛♥❞ ✈♦❧✉♠❡s ♦❢ r❡❣✉❧❛r ♣♦❧②❤❡❞r❛✳ ❋✐♥❛❧❧②✱ ✇❡ ♣r❡s❡♥t s♦♠❡ ♠❛t❤❡♠❛t✐❝❛❧ s♦❢t✇❛r❡ t❤❛t ❝❛♥ ❜❡ ✉s❡❞ ❜② st✉❞❡♥ts ❛♥❞ ♠❛t❤❡♠❛t✐❝s t❡❛❝❤❡rs ✐♥ t❤❡ s♣❛t✐❛❧ ❣❡♦♠❡tr② ❝❧❛ss❡s ❛s ❛✉①✐❧✐❛r② ♠❛t❡r✐❛❧ ✐♥ t❤❡ t❡❛❝❤✐♥❣ ❛♥❞ ❧❡❛r♥✐♥❣ ♦❢ ❑❡②✇♦r❞st❤✐s s✉❜❥❡❝t ✐♥ t❤❡ ❝❧❛ssr♦♦♠✳ ✿ s♣❛t✐❛❧ ❣❡♦♠❡tr②✱ r❡❣✉❧❛r ♣♦❧②❤❡❞r❛✱ ❊✉❧❡r✬s ❚❤❡♦r❡♠✱ ▼❛t❤❡♠❛t✐❝s ❙♦❢t✇❛r❡✳ ✈✐✐

(8) ❙✉♠ár✐♦ ■♥tr♦❞✉çã♦ ①✐✐✐ ✶ ❈♦♥❝❡✐t♦ ❍✐stór✐❝♦ ✶ ✷ ❖s ♣♦❧✐❡❞r♦s ♥♦s ❧✐✈r♦s ❞✐❞át✐❝♦s ✺ ✷✳✶ P♦❧✐❡❞r♦s ❝♦♥✈❡①♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✷✳✷ ❊❧❡♠❡♥t♦s ❞♦s ♣♦❧✐❡❞r♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✷✳✸ ❘❡❧❛çõ❡s ❡♥tr❡ ♦s ❊❧❡♠❡♥t♦s ❞♦s P♦❧✐❡❞r♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✷✳✸✳✶ ❋❛❝❡s ❡ ❆r❡st❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷✳✸✳✷ ❱ért✐❝❡s ❡ ❆r❡st❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷✳✸✳✸ ❘❡❧❛çã♦ ❞❡ ❊✉❧❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✸ ❆ ❘❡❧❛çã♦ ❞❡ ❊✉❧❡r ✶✹ ✹ P♦❧✐❡❞r♦s r❡❣✉❧❛r❡s ✶✼ ✹✳✶ ❊①✐stê♥❝✐❛ ❞❡ ❝✐♥❝♦ ♣♦❧✐❡❞r♦s r❡❣✉❧❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✹✳✷ ➪r❡❛s ❡ ❱♦❧✉♠❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✹✳✷✳✶ ➪r❡❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✹✳✷✳✷ ❱♦❧✉♠❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✹✳✸ ❖✉tr♦s P♦❧✐❡❞r♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✺ Pr♦s♣♦st❛s ❞❡ ❙♦❢t✇❛r❡s ✹✵ ✺✳✶ ❈❛❜r✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✺✳✷ P♦❧② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✺✳✸ ❖✉tr♦s ❙♦❢t✇❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✹✻ ✈✐✐✐

(9) ▲✐st❛ ❞❡ ❋✐❣✉r❛s ✶✳✶ ❋r❛❣♠❡♥t♦ ❞♦ ♣❛♣✐r♦ ❞❡ ▼♦s❝♦✈♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷ ❙í♠❜♦❧♦ ❞❛ ❡s❝♦❧❛ P✐t❛❣ór✐❝❛✱ ♦ ♣❡♥t❛❣r❛♠❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸ ❙ó❧✐❞♦s P❧❛tô♥✐❝♦s ❡ s❡✉s r❡s♣❡❝t✐✈♦s ❡❧❡♠❡♥t♦s ❞❛ ♥❛t✉r❡③❛✳ ✳ ✳ ✳ ✳ ✳ ✶✳✹ P✐râ♠✐❞❡s ❞♦ ❊❣✐t♦ ❡ ❛ P✐râ♠✐❞❡ ❞♦ ♠✉s❡✉ ❞♦ ▲♦✉✈r❡✱ ❢♦r♠❛s ♣♦❧✐é✲ ❞r✐❝❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✸ ✹ ✹ ✷✳✶ ❋✐❣✉r❛s P♦❧✐é❞r✐❝❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷ ❋✐❣✉r❛s ♥ã♦ P♦❧✐é❞r✐❝❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✸ P♦❧✐❡❞r♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✹ P♦❧✐❡❞r♦s ♥ã♦ ❝♦♥✈❡①♦ ✭à ❡sq✉❡r❞❛✮ ❡ P♦❧✐❡❞r♦ ❝♦♥✈❡①♦ ✭à ❞✐r❡✐t❛✮✳ ✳ ✳ ✷✳✺ P♦❧í❣♦♥♦s ♥♦ ♣❧❛♥♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✻ P♦❧✐❡❞r♦ P ♣❧❛♥✐✜❝❛❞♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✼ ❱✐st❛ s✉♣❡r✐♦r ❞♦ ♣♦❧✐❡❞r♦ P ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✻ ✼ ✽ ✽ ✾ ✾ ✹✳✶ ❚❡tr❛❡❞r♦ ✭❡sq✉❡r❞❛✮ ❡ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ♣❧❛♥✐✜❝❛❞❛ ✭❞✐r❡✐t❛✮✳ ✳ ✳ ✳ ✳ ✶✾ ✹✳✷ ❖❝t❛❡❞r♦ ✭❡sq✉❡r❞❛✮ ❡ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ♣❧❛♥✐✜❝❛❞❛ ✭❞✐r❡✐t❛✮✳ ✳ ✳ ✳ ✳ ✷✵ ✹✳✸ ■❝♦s❛❡❞r♦ ✭❡sq✉❡r❞❛✮ ❡ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ♣❧❛♥✐✜❝❛❞❛ ✭❞✐r❡✐t❛✮✳ ✳ ✳ ✳ ✳ ✷✵ ✹✳✹ ❍❡①❛❡❞r♦ ✭❡sq✉❡r❞❛✮ ❡ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ♣❧❛♥✐✜❝❛❞❛ ✭❞✐r❡✐t❛✮✳ ✳ ✳ ✳ ✳ ✷✶ ✹✳✺ ❉♦❞❡❝❛❡❞r♦ ✭❡sq✉❡r❞❛✮ ❡ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ♣❧❛♥✐✜❝❛❞❛ ✭❞✐r❡✐t❛✮✳ ✳ ✳ ✷✷ ✹✳✻ ❚r✐â♥❣✉❧♦ DEF ❞❡ ❧❛❞♦ α✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✹✳✼ ◗✉❛❞r❛❞♦ ❞❡ ❧❛❞♦ α✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✹✳✽ P❡♥tá❣♦♥♦ JKLMN ❞❡ ❧❛❞♦ α✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✹✳✾ ❚❡tr❛❡❞r♦ ❞❡ ❛r❡st❛ α✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✹✳✶✵ ❚r✐â♥❣✉❧♦ ❞❛ ❜❛s❡ ❞♦ t❡tr❛❡❞r♦ ❞❡ ❛r❡st❛ α✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✹✳✶✶ ❖❝t❛❡❞r♦ ❞❡ ❛r❡st❛ α✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✹✳✶✷ ❉♦❞❡❝❛❡❞r♦ ❞❡❝♦♠♣♦st♦ ❡♠ ♦✉tr♦s ♣♦❧í❣♦♥♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✹✳✶✸ ❘❡t❛ s s❡❝❝✐♦♥❛♥❞♦ ♦ ♣❡♥tá❣♦♥♦ ❏❑▲▼◆✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✹✳✶✹ Pr✐s♠❛ r❡t♦ ♥♦ ♠❡✐♦ ❡ ❛ ❥✉♥çã♦ ❞♦s ♣♦❧✐❡❞r♦s ❞❛s ❧❛t❡r❛✐s ❢♦r♠❛♠ ✉♠❛ ♣✐râ♠✐❞❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✹✳✶✺ ❉✐✈✐❞✐♥❞♦ ♦ ✐❝♦s❛❡❞r♦ ❡♠ ✷✵ ♣✐râ♠✐❞❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✹✳✶✻ P♦❧✐❣♦♥❛❧ KP QRMS ❡ ♣❡♥tá❣♦♥♦ r❡❣✉❧❛r JKLMN✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✹✳✶✼ ❉✐❛❣♦♥❛❧ d ❞♦ ✐❝♦s❛❡❞r♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✐①

(10) ✹✳✶✽ ❆❧t✉r❛ CG ❞❡ ✉♠❛ ❞❛s ✷✵ ♣✐râ♠✐❞❡s q✉❡ ❢♦r♠❛♠ ♦ ✐❝♦s❛❡❞r♦ ❞❡ ❛r❡st❛ α✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✹✳✶✾ P♦❧✐❡❞r♦s ❙❡♠✐rr❡❣✉❧❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✹✳✷✵ P✐râ♠✐❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✹✳✷✶ Pr✐s♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✺✳✶ ❏❛♥❡❧❛s ❞♦ ❈❛❜r✐ ✸❉✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✺✳✷ P❧❛♥✐✜❝❛♥❞♦ ♦ ❝✉❜♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✺✳✸ ❇❛rr❛ ❞❡ ❢❡rr❛♠❡♥t❛s ❞♦ P♦❧②✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✺✳✹ P❧❛♥✐✜❝❛♥❞♦ ♦ ✐❝♦s❛❡❞r♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷

(11) ▲✐st❛ ❞❡ ❚❛❜❡❧❛s ✹✳✶ ✹✳✷ ✹✳✸ P♦❧✐❡❞r♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ➪r❡❛ ❡ ✈♦❧✉♠❡ ❞❡ ✉♠❛ ♣✐râ♠✐❞❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ➪r❡❛ ❡ ✈♦❧✉♠❡ ❞❡ ✉♠ ♣r✐s♠❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ①✐

(12) ◆♦t❛çõ❡s ◆♦t❛çõ❡s ●❡r❛✐s ❼ P é ✉♠ ♣♦❧✐❡❞r♦ ❝♦♥✈❡①♦ q✉❛❧q✉❡r✳ ❼ A é ♦ ♥ú♠❡r♦ ❞❡ ❛r❡st❛s ❞❡ ✉♠ ♣♦❧✐❡❞r♦ ❝♦♥✈❡①♦ P ✳ ❼ F Pé ♦ ♥ú♠❡r♦ ❞❡ ❢❛❝❡s ❞❡ ✉♠ ♣♦❧✐❡❞r♦ ❝♦♥✈❡①♦ ✳ ❼ V Pé ♦ ♥ú♠❡r♦ ❞❡ ✈ért✐❝❡s ❞❡ ✉♠ ♣♦❧✐❡❞r♦ ❝♦♥✈❡①♦ ✳ ❼ Fn é ❛ ♥✲és✐♠❛ ❢❛❝❡ ❞❡ ✉♠ ♣♦❧✐❡❞r♦ ❝♦♥✈❡①♦✳ ❼ nk é ♦ ❦✲és✐♠♦ ❧❛❞♦ ❞❡ ✉♠❛ ❢❛❝❡✳ ❼ Vp é ❛ ♣✲és✐♠❛ ❛r❡st❛ q✉❡ ✐♥❝✐❞❡ ♥✉♠ ✈ért✐❝❡✳ ❼ S é ❛ ár❡❛ ❞❡ ✉♠❛ s✉♣❡r❢í❝✐❡ ❞❛s ❢❛❝❡s ❞❡ ✉♠ ♣♦❧✐❡❞r♦✳ ❼ Sb é ❛ ár❡❛ ❞❛ s✉♣❡r❢í❝✐❡ ❞❛ ❜❛s❡ ❞❡ ✉♠ ♣♦❧✐❡❞r♦✳ S❼ l é ❛ ár❡❛ ❞❛ s✉♣❡r❢í❝✐❡ ❞❛s ❢❛❝❡s ❧❛t❡r❛✐s ❞❡ ✉♠ ♣♦❧✐❡❞r♦✳ ❼ V é ♦ ✈♦❧✉♠❡ ❞❡ ✉♠ ♣♦❧✐❡❞r♦ r❡❣✉❧❛r✳ ❼ Vpir é ♦ ✈♦❧✉♠❡ ❞❡ ✉♠❛ ♣✐râ♠✐❞❡✳ ❼ d é ❛ ❞✐❛❣♦♥❛❧ ♠❛✐♦r ❞♦ ✐❝♦s❛❡❞r♦✳ ❼ h ❛❧t✉r❛ ❞❡ ✉♠❛ ❢❛❝❡ ♦✉ ❞❡ ✉♠ ♣♦❧✐❡❞r♦✳ ❼ α é ❛r❡st❛ ❞❡ ✉♠ ♣♦❧✐❡❞r♦ r❡❣✉❧❛r✳ ①✐✐

(13) ■♥tr♦❞✉çã♦ ❊st❡ tr❛❜❛❧❤♦ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ ❛♥❛❧✐s❛r ❛ ❢♦r♠❛ ❞❡ ❝♦♠♦ ♦s ♣♦❧✐❡❞r♦s r❡❣✉❧❛r❡s sã♦ ✈✐st♦s ♣♦r ❛❧❣✉♥s ❛✉t♦r❡s ❞❡ ❧✐✈r♦s ❞✐❞át✐❝♦s ♣❛r❛ ♦ ❊♥s✐♥♦ ▼é❞✐♦✳ Pr✐♠❡✐r❛♠❡♥t❡ ❢♦✐ r❡❛❧✐③❛❞❛ ✉♠❛ ❜r❡✈❡ ♣❛ss❛❣❡♠ ♣♦r ❛❧❣✉♥s ❢❛t♦s q✉❡ ♣♦❞❡♠ t❡r ❞❛❞♦ í♥✐❝✐♦ ♦✉ ♠❛r❝❛r❛♠ ❛ ❤✐stór✐❛ ❞❛ ●❡♦♠❡tr✐❛ ❊s♣❛❝✐❛❧✱ ♥❡ss❡ ♠❡s♠♦ ❝❛♣ít✉❧♦ ✈❡r❡♠♦s ♦s ♣r✐♥❝✐♣❛✐s ❣❡ô♠❡tr❛s q✉❡ ❞❡❞✐❝❛r❛♠✲s❡ ❛♦ ❡st✉❞♦ ❞♦s só❧✐❞♦s ❣❡♦♠étr✐❝♦s ❞❡♥tr❡ ❡❧❡s✱ ❞❡st❛❝❛✲s❡ P❧❛tã♦ q✉❡ t❡♠ s❡✉ ♥♦♠❡ ❛ss♦❝✐❛❞♦ ❛♦s ♣♦❧✐❡❞r♦s r❡❣✉❧❛r❡s✳ ◆♦ ❝❛♣ít✉❧♦ ❞♦✐s ✐♥✐❝✐❛♠♦s ❝♦♠ ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s ♣❛r❛ q✉❛❧q✉❡r ♣♦❧✐❡❞r♦ ❡♠ ❛❧❣✉♥s ❧✐✈r♦s ❞✐❞át✐❝♦s✱ ❞❡♣♦✐s ❞✐✈✐❞✐♠♦s ♦s ♣♦❧✐❡❞r♦s ❡♠ ❞♦✐s ❣r✉♣♦s✿ ♦s ❝♦♥✈❡①♦s ❡ ♦s ♥ã♦ ❝♦♥✈❡①♦s ✭❝ô♥❝❛✈♦s✮✳ ❖ ♣ró①✐♠♦ ♣❛sss♦ ❢♦✐ ✐❞❡♥t✐✜❝❛r ❡ ❝❧❛ss✐✜❝❛r ♦s ❡❧❡♠❡♥t♦s q✉❡ ❝♦♠♣õ❡♠ ✉♠ ♣♦❧✐❡❞r♦✿ ❢❛❝❡s✱ ✈ért✐❝❡s ❡ ❛r❡st❛s✳ ❈♦♠ ❡ss❡s ❡❧❡♠❡♥t♦s ❞❡✜♥✐❞♦s ♣♦❞❡♠♦s ❡s❝r❡✈❡r ❞✉❛s r❡❧❛çõ❡s ❡♥tr❡ ❡❧❡s✳ ❖ ♣ró①✐♠♦ ❝❛♣ít✉❧♦ ✉t✐❧✐③❛♠♦s ❛s r❡❧❛çõ❡s ❡♥tr❡ ♦s ❡❧❡♠❡♥t♦s ❞♦s ♣♦❧✐❡❞r♦s ♣❛r❛ ❞❡♠♦♥str❛r♠♦s ♦ ❢❛♠♦s♦ t❡♦r❡♠❛ ❞❡ ❊✉❧❡r✱ q✉❡ r❛r❛♠❡♥t❡ é ❞❡♠♦♥str❛❞♦ ♥♦s ❧✐✈r♦s ❞✐❞át✐❝♦s✱ ♠❛s ❜❛st❛♥t❡ tr❛❜❛❧❤❛❞♦ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦ q✉❛♥❞♦ s❡ ❡st✉❞❛ ♣♦❧✐❡❞r♦s✳ ◆♦ ❝❛♣ít✉❧♦ q✉❛tr♦ ❞❡✜♥✐♠♦s ♣♦❧✐❡❞r♦s r❡❣✉❧❛r❡s ✉t✐❧✐③❛♥❞♦✲s❡ ❞❡ ❞✉❛s ❞❡✜♥✐çõ❡s ❡ ❝♦♠ ♦ t❡♦r❡♠❛ ❞❡ ❊✉❧❡r ❞❡♠♦♥str❛❞♦ ♣r♦✈❛♠♦s ❛ ❡①✐stê♥❝✐❛ ❞❡ ❛♣❡♥❛s ❝✐♥❝♦ ♣♦✲ ❧✐❡❞r♦s r❡❣✉❧❛r❡s ♦s só❧✐❞♦s ❞❡ P❧❛tã♦✱ ❛♣r❡s❡♥t❛✲s❡ t❛♠❜é♠ ♦s ✈❛❧♦r❡s ❞❛s ár❡❛s ❞❛s s✉♣❡r❢í❝✐❡s ❡ ✈♦❧✉♠❡s ❞❡ss❡s ❝✐♥❝♦ ♣♦❧✐❡❞r♦s ❡♠ ❢✉♥çã♦ ❞❡ s✉❛ ❛r❡st❛ ❡ á♣♦t❡♠❛✳ ◆♦ ✜♠ ❞♦ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛♠♦s ✉♠ ❜r❡✈❡ r❡s✉♠♦ ❞❡ ♦✉tr♦s ♣♦❧✐❡❞r♦s ♥ã♦ r❡❣✉❧❛r❡s ♠❛s q✉❡ ❛♣r❡s❡♥t❛♠✲s❡ ❞❡ ❢♦r♠❛ ❞❡st❛❝❛❞❛ ♥♦s ❧✐✈r♦s ❞✐❞át✐❝♦s✳ ❋✐♥❛❧✐③❛♠♦s ♦ tr❛❜❛❧❤♦ ❝♦♠ ✉♠❛ ♣r♦♣♦st❛ ❞❡ ✉s♦ ❞♦s s♦❢t✇❛r❡s ❈❛❜r✐ ❡ P♦❧② ♥❛s ❛♣r❡s❡♥t❛çõ❡s ❞❡ ❝♦♥t❡ú❞♦s ❡♥✈♦❧✈❡♥❞♦ ♣♦❧✐❡❞r♦s ♣❛r❛ ♠❡❧❤♦r ✜①❛çã♦ ❡ ❝♦♠♣r❡❡♥sã♦ ❞♦ ❛ss✉♥t♦✳ ①✐✐✐

(14) ❈❛♣ít✉❧♦ ✶ ❈♦♥❝❡✐t♦ ❍✐stór✐❝♦ ❆ ♠❛t❡♠át✐❝❛ ❞❡s♣❡rt❛ ♦ ✐♥t❡r❡ss❡ ❞❛s ♣❡ss♦❛s ❞❡s❞❡ ❛ ❛♥t✐❣✉✐❞❛❞❡✱ ❡❧❛ é ✉♠❛ ❞❛s ❝✐ê♥❝✐❛s ♠❛✐s ❛♥t✐❣❛s ❡ s✉❛ ♦r✐❣❡♠ ♥ã♦ t❡♠ ❞❛t❛ ✜①❛✱ ❛ ❞❛t❛ ♣r❡❝✐s❛ ❞❡ q✉❛♥❞♦ ♦❝♦rr❡r❛♠ ♦s ♣r✐♠❡✐r♦s ❝á❧❝✉❧♦s q✉❡ ❡♥✈♦❧✈❡r❛♠ ✉♠ r❛❝✐♦❝í♥✐♦ ❣❡♦♠étr✐❝♦ t❛♠❜é♠ ♥ã♦ é ♣r❡❝✐s❛✱ ❛❝r❡❞✐t❛✲s❡ q✉❡ ♥❛ ♣ré✲❤✐stór✐❛ ♦ ❤♦♠❡♠ ❥á ❞❡s❡♥✈♦❧✈✐❛ ✉♠ r❛❝✐♦❝í♥✐♦ ♠❛t❡♠át✐❝♦✱ ❝♦♥t✉❞♦ ♥ã♦ ❤❛✈✐❛ r❡❣✐str♦s ❞❡ t❛✐s ❢❡✐t♦s✳ P♦❞❡♠♦s ❝✐t❛r ❝♦♠♦ ✉♠ ❞♦s ♣r✐♠❡✐r♦s t❡①t♦s ♠❛t❡♠át✐❝♦s ✭♣r♦✈❡♥✐❡♥t❡s ✷✵✵✵ ❛✳❈✳✮ ♦ ♣❛♣✐r♦ ♠❛t❡♠át✐❝♦ ❞❡ ❘❤✐♥❞ ✭♠❛t❡♠át✐❝❛ ❡❣í♣❝✐❛✱ ❝❡r❝❛ ❞❡ ✷✵✵✵ ✲ ✶✽✵✵ ❛✳❈✳✮ ❡ ♦ P❛♣✐r♦ ▼❛t❡♠át✐❝♦ ❞❡ ▼♦s❝♦✈♦ ♦✉ ▼♦s❝♦✉ ✭♠❛t❡♠át✐❝❛ ❡❣í♣❝✐❛✱ ❝❡r❝❛ ❞❡ ✶✽✾✵ ❛✳❈✳✮✳ ❖ ♣❛♣✐r♦ ❞❡ ▼♦s❝♦✈♦ q✉❡ ❢♦✐ ❡s❝r✐t♦ ♣♦r ✉♠ ❡s❝r✐❜❛ ❞❡s❝♦♥❤❡❝✐❞♦ ❝♦♥té♠ ✈✐♥t❡ ❡ ❝✐♥❝♦ ❡①❡♠♣❧♦s✱ q✉❛s❡ t♦❞♦s ❞❛ ✈✐❞❛ ♣rát✐❝❛ ❡ ♥ã♦ ❞✐❢❡r❡♥❝✐❛♥❞♦ ♠✉✐t♦ ❞♦ ♣❛♣✐r♦ ❘❤✐♥❞✳ ❆ss♦❝✐❛❞♦ ❛♦ ♣r♦❜❧❡♠❛ ✶✹ ❞♦ ♣❛♣✐r♦ ❞❡ ▼♦s❝♦✈♦ ❤á ✉♠❛ ✜❣✉r❛ q✉❡ é ♣❛r❡❝✐❞❛ ❝♦♠ ✉♠ tr❛♣é③✐♦✱ ♣♦ré♠ ♦s ❝á❧❝✉❧♦s ❛ss♦❝✐❛❞♦s ❛ ❡❧❛ ♠♦str❛♠ q✉❡ ♦ q✉❡ s❡ r❡♣r❡s❡♥t❛ é ✉♠ tr♦♥❝♦ ❞❡ ✉♠❛ ♣✐râ♠✐❞❡ ✭✈❡❥❛ ❬✺❪ ♣á❣✐♥❛ ✶✹✮✳ ❊ss❡ t❛❧✈❡③ s❡❥❛ ♦ ♣r✐♠❡✐r♦ r❡❣✐str♦ ❞❡ ✉♠ ❝á❧❝✉❧♦ ❡♥✈♦❧✈❡♥❞♦ ●❡♦♠❡tr✐❛ ❊s♣❛❝✐❛❧✳ ❋✐❣✉r❛ ✶✳✶✿ ❋r❛❣♠❡♥t♦ ❞♦ ♣❛♣✐r♦ ❞❡ ▼♦s❝♦✈♦✳ ❚♦❞❛✈✐❛ ♣♦❞❡rí❛♠♦s t❡r r❡❣✐str♦s ❛♥t❡r✐♦r❡s ❛ ❡ss❡s ❡♥❝♦♥tr❛❞♦s ♥♦ ♣❛♣✐r♦ ▼♦s✲ ❝♦✈♦✱ ❝♦♥t✉❞♦ ❛ ❛✉sê♥❝✐❛ ❞❡ss❡s r❡❣✐str♦s t♦r♥♦✉ ❛ ♦r✐❣❡♠ ❞❛ ●❡♦♠❡tr✐❛ ✐♥❞❡t❡r♠✐✲ ✶

(15) ♥❛❞❛✱ ♣♦ré♠ ♣♦❞❡♠♦s ❞❡st❛❝❛r ❛❧❣✉♠❛s ✉t✐❧✐③❛çõ❡s ❞❡ ❝♦♥❝❡✐t♦s ❞❡ ❛❧❣✉♥s ♣♦✈♦s ♥❛ ❛♥t✐❣✉✐❞❛❞❡✱ ❡ q✉❡ ❛t✉❛❧♠❡♥t❡ ❛✐♥❞❛ sã♦ út❡✐s ♦✉ s❡r✈❡♠ ❞❡ ✐♥s♣✐r❛çã♦ ♦✉ ❜❛s❡ ❡♠ ♥♦✈❛s ♣r♦♣♦st❛s ❞❡ ✐❞❡✐❛s ❣❡♦♠étr✐❝❛s❀ ❝♦♠♦ ♦s ❡❣í♣❝✐♦s ❡ ♦ ✉s♦ ❞❡ ár❡❛s ❞❡ t❡rr❡♥♦s ♥❛ ❛❣r✐❝✉❧t✉r❛ ❡ s✉❛s ❝♦♥str✉çõ❡s t❡♥❞♦ ❛s ♣✐râ♠✐❞❡s ❝♦♠♦ ❡①❡♠♣❧♦✱ ♣♦❞❡♠♦s t❛♠✲ ❜é♠ ❞❡st❛❝❛r ♦✉tr♦s ♣♦✈♦s ❛♥t✐❣♦s ❝♦♠♦ ♦s ❝❤✐♥❡s❡s ❡ s✉❛ ❣r❛♥❞❡ ♠✉r❛❧❤❛✱ ♣♦ré♠ ❢♦✐ ♥❛ ●ré❝✐❛✱ q✉❡ ❛ ●❡♦♠❡tr✐❛ t❡✈❡ s❡✉ ♠❛✐♦r ❞❡s❡♥✈♦❧✈✐♠❡♥t♦✳ ❖s ❣r❡❣♦s ♣❡r❝❡❜❡r❛♠ ❡♠ s❡✉s ❡st✉❞♦s q✉❡ ♦s ❡❣í♣❝✐♦s ❢♦r❛♠ ❝❛♣❛③❡s ❞❡ r❡❛❧✐③❛r ❝á❧❝✉❧♦s ❡ ♠❡❞✐❞❛s ❞❡ ❞✐♠❡♥s✐♦♥❛♠❡♥t♦ ❞❛ t❡rr❛✱ ❡❧❡s ♣r♦❝✉r❛r❛♠ ❞❡♠♦♥str❛r ❧❡✐s ❛❝❡r❝❛ ❞♦ ❡s♣❛ç♦✱ ❞❛í ♦ ♥♦♠❡ ●❊❖▼❊❚❘■❆✱ ❣❡♦ ✲ t❡rr❛ ❡ ♠❡tr✐❛ ✲ ♠❡❞✐❞❛✳ ❈♦♠ ❧♦❝❛❧✐③❛çã♦ ♣r✐✈✐❧❡❣✐❛❞❛ ❡♥tr❡ ♦s ♠❛r❡s ❊❣❡✉ ❡ ❏ô♥✐♦ ❡ ❝♦❧ô♥✐❛s q✉❡ s❡ ❡♥✲ ❝♦♥tr❛✈❛♠ ♥❛s ♠❛r❣❡♥s ❞♦s ♠❛r❡s ◆❡❣r♦ ❡ ▼❡❞✐t❡rrâ♥❡♦✱ ❛❧é♠ ❞❛ ót✐♠❛ ❧♦❝❛❧✐③❛çã♦ ❣❡♦❣rá✜❝❛ q✉❡ ❣❛r❛♥t✐❛ ♣♦❞❡r❡♠ ✈✐❛❥❛r ❛♦s ❣r❛♥❞❡s ❝❡♥tr♦s ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞❛ é♣♦❝❛ ❥✉♥t♦ ❛ ✉♠ ❡s♣ír✐t♦ ♦✉s❛❞♦ ❡ ✐♠❛❣✐♥❛t✐✈♦ ❢❡③ ❝♦♠ q✉❡ ♦s ❛❞q✉✐r✐ss❡♠ ✐♥❢♦r✲ ♠❛çõ❡s ❞❡ ♣r✐♠❡✐r❛ ♠ã♦ s♦❜r❡ ♠❛t❡♠át✐❝❛ ❡ ❛str♦♥♦♠✐❛✳ ❊ ❢♦✐ ♥♦ ❊❣✐t♦ ❞✐③✲s❡ q✉❡ ❛♣r❡♥❞❡r❛♠ ❣❡♦♠❡tr✐❛✳ ❋♦r❛♠ ♠✉✐t♦s ♦s ❣❡ô♠❡tr❛s ❣r❡❣♦s q✉❡ ♣♦❞❡♠♦s ❞❡st❛❝❛r ❡♥tr❡ ❡❧❡s ❚❛❧❡s✱ P✐tá❣♦r❛s✱ ❊✉❝❧✐❞❡s ❡ P❧❛tã♦✳ ❚❛❧❡s ❢r❡q✉❡♥t❡♠❡♥t❡ s❛✉❞❛❞♦ ❝♦♠♦ ♦ ♣r✐♠❡✐r♦ ♠❛t❡♠át✐❝♦ ✈❡r❞❛❞❡✐r♦ ✲ ♦r✐❣✐♥❛✲ ❞♦r ❞❛ ♦r❣❛♥✐③❛çã♦ ❞❡❞✉t✐✈❛ ❞❛ ❣❡♦♠❡tr✐❛✳ ◆ã♦ ❤á ❞♦❝✉♠❡♥t♦ q✉❡ ♣r♦✈❡ t❛❧ ❢❡✐t♦✱ ♦ ♠❛✐s ♣ró①✐♠♦ q✉❡ ♣♦❞❡♠♦s ❝❤❡❣❛r ❛ ✐ss♦ ❞✐❣♥♦ ❞❡ ❝♦♥✜❛♥ç❛ é ✉♠❛ ♠❡♥çã♦ ✶✵✵✵ ❛♥♦s ❞❡♣♦✐s ❞♦ t❡♠♣♦ ❞❡ ❚❛❧❡s✳ ❯♠ ❞✐s❝í♣✉❧♦ ❞❡ ❆r✐stót❡❧❡s ❝❤❛♠❛♥❞♦ ❊✉❞❡♠♦ ❡s❝r❡✈❡✉ ✉♠❛ ❤✐stór✐❛ ❞❛ ♠❛t❡♠át✐❝❛✱ ❡ss❛ s❡ ♣❡r❞❡✉ ♣♦ré♠ ❛❧❣✉é♠ r❡s✉♠✐✉ ♣❛rt❡ ❞❡❧❛✱ ❡ss❡ r❡s✉♠♦ t❛♠❜é♠ ♣❡r❞❡✉✲s❡✱ ♣♦ré♠ Pr♦❝❧✉s ✜❧ós♦❢♦ ♥❡♦✲♣❧❛tô♥✐❝♦ ❡s❝r❡✈❡✉ ♣❛rt❡ ❞❛s ✐♥❢♦r♠❛çõ❡s ❞♦ s✉♠ár✐♦ ❞❡ ❊✉❞❡♠♦s✱ ♥❡❧❡ ❞✐③ ♦ s❡❣✉✐♥t❡ s♦❜r❡ ❚❛❧❡s✳ ✭✈❡❥❛ ❬✺❪ ♣á❣✐♥❛s ✸✹✲✸✺✮✳ ✳✳✳Pr✐♠❡✐r♦ ❢♦✐ ❛♦ ❊❣✐t♦ ❡ ❞❡ ❧á ✐♥tr♦❞✉③✐✉ ❡ss❡ ❡st✉❞♦ ♥❛ ●ré❝✐❛✳ ❉❡s❝♦❜r✐✉ ♠✉✐t❛s ♣r♦♣♦s✐çõ❡s ❡❧❡ ♣ró♣r✐♦ ❡ ✐♥str✉✐✉ s❡✉s s✉❝❡ss♦r❡s ♥♦s ♣rí♥❝✐♣✐♦s q✉❡ r❡❣❡♠ ♠✉✐t❛s ♦✉tr❛s✳✳✳ ❏á P✐tá❣♦r❛s✱ ❛❧❣✉♥s ❞✐③❡♠ ❞✐s❝í♣✉❧♦ ❞❡ ❚❛❧❡s✱ t❡✈❡ t❛♠❜é♠ s✉❛s ❝♦♥tr✐❜✉✐çõ❡s ♥❛ ❣❡♦♠❡tr✐❛✳ ❋✉♥❞❛❞♦r ❞❛ ❢❛♠♦s❛ ❡s❝♦❧❛ ♣✐t❛❣ór✐❝❛ ❡ ❞♦ ❢❛♠♦s♦ t❡♦r❡♠❛ ✱ ♣r♦✈❛❧✲ ✈❡♠❡♥t❡ ❞❡ ❡st✉❞♦s ♥❛s ♣✐râ♠✐❞❡s ❞♦ ❊❣✐t♦✱ ♥♦ s✉♠ár✐♦ ❞❡ ❊✉❞❡♠♦✲Pr♦❝❧✉s ❧❤❡ é ❛tr✐❜✉í❞❛ ❛ ❝♦♥str✉çã♦ ❞❡ ✧✜❣✉r❛s ❝ós♠✐❝❛s✧ ✭✐st♦ é só❧✐❞♦s r❡❣✉❧❛r❡s✮✱ ♣♦ré♠ ❡①✐s✲ t❡♠ ❛❧❣✉♠❛s ❞ú✈✐❞❛s✳ ❖ ❤❡①❛❡❞r♦✱ ♦ ♦❝t❛❡❞r♦ ❡ ♦ ❞♦❞❡❝❛❡❞r♦ ♣♦❞✐❛♠ t❡r s✐❞♦ ♦❜s❡r✈❛❞♦s ❡♠ ❝r✐st❛✐s✱ ❝♦♠♦ ♦ ❞❛ ♣✐r✐t❛ ✭❞✐ss✉❧✜t♦ ❞❡ ❢❡rr♦✮✱ ♠❛s ❡♠ ❖s ❊❧❡♠❡♥t♦s ❳■■■ ❡stá ❞✐t♦ q✉❡ ♦s ♣✐t❛❣ór✐❝♦s só ❝♦♥❤❡❝✐❛♠ três ♣♦❧✐❡❞r♦s r❡❣✉❧❛r❡s❀ ♦ t❡tr❛❡❞r♦✱ ♦ ❤❡①❛❡❞r♦ ❡ ♦ ❞♦❞❡❝❛❡❞r♦✳ ❙♦❜r❡ ♦ ú❧t✐♠♦ ♥ã♦ s❡r✐❛ ✐♠♣r♦✈á✈❡❧ ♣♦✐s ❡❧❡s ❝♦♥❤❡❝✐❛♠ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦ ♣❡♥tá❣♦♥♦ r❡❣✉❧❛r✳ ✭❖ ♣❡♥t❛❣r❛♠❛✱ sí♠❜♦❧♦ ❡s♣❡❝✐❛❧ ❞❛ ❡s❝♦❧❛ ♣✐t❛❣ór✐❝❛✱ ❢♦r♠❛❞❛ tr❛ç❛♥❞♦ ❛s ❞✐❛❣♦♥❛✐s ❞❡ ✉♠ ♣❡♥tá❣♦♥♦ q✉❡ é ❢❛❝❡ ❞♦ ❞♦❞❡❝❛❡❞r♦✮✳ ✭✈❡❥❛ ❬✺❪ ♣á❣✐♥❛ ✸✼✮✳ ✷

(16) ❋✐❣✉r❛ ✶✳✷✿ ❙í♠❜♦❧♦ ❞❛ ❡s❝♦❧❛ P✐t❛❣ór✐❝❛✱ ♦ ♣❡♥t❛❣r❛♠❛✳ ❊✉❝❧✐❞❡s ❞❡ ❆❧❡①❛♥❞r✐❛✱ ❛ss✐♠ ❝♦♥❤❡❝✐❞♦ ♣♦r t❡r ❡♥s✐♥❛❞♦ ♠❛t❡♠át✐❝❛ ♥❛ ❝✐❞❛❞❡ ❞❡ ❆❧❡①❛♥❞r✐❛✱ ❢♦✐ ♦✉tr♦ ❣r❛♥❞❡ ❣❡ô♠❡tr❛ ❣r❡❣♦✱ ❡s❝r✐t♦r ❞❡ ❞✐✈❡rs❛s ♦❜r❛s ✭❝✐♥❝♦ s♦❜r❡✈✐✈❡♠ ❛té ❤♦❥❡✮ ❞❡♥tr❡ ❡❧❛s ❞❡st❛❝❛✲s❡ ✉♠❛ ♦❜r❛ ❞♦ ✉♥✐✈❡rs♦ ♠❛t❡♠át✐❝♦ q✉❡ ❛té ❤♦❥❡ t❡♠ ❣r❛♥❞❡ ❝♦♥tr✐❜✉✐çã♦ ❡ t❛❧✈❡③ s❡❥❛ ✉♠❛ ❞❛s ♠❛✐♦r❡s ♦❜r❛s ♣✉❜❧✐❝❛❞❛s ✭♣❛r❛ ❛❧❣✉♥s ♠❛t❡♠át✐❝♦s ❛ ♠❛✐♦r✮ é ❖s ❊❧❡♠❡♥t♦s ❞❡ ❊✉❝❧✐❞❡s✱ s❡❣✉♥❞♦ ➪✈✐❧❛ ❬✶❪✿ ✧◆ã♦ s❛❜❡♠♦s s❡ ❊✉❝❧✐❞❡s ❡s❝r❡✈❡✉ ❖s ❊❧❡♠❡♥t♦s ♣❛r❛ ✉s♦ ♥♦ ❡♥s✐♥♦✱ ♦✉ ❛♣❡♥❛s ♣❛r❛ r❡✉♥✐r ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ♠❛t❡♠át✐❝♦ ❞❛ é♣♦❝❛✳ ◆❛q✉❡❧❡ t❡♠♣♦ ♥ã♦ ❤❛✈✐❛ ❛ ♣r❡♦❝✉♣❛çã♦ ♣❡❞❛❣ó❣✐❝❛ ❞♦s ❞✐❛s ❞❡ ❤♦❥❡✱ ❞❡ s♦rt❡ q✉❡ ❊✉❝❧✐❞❡s ❛❧❝❛♥ç♦✉ ♦s ❞♦✐s ♦❜❥❡t✐✈♦s❀ ❡ ♦s ❊❧❡♠❡♥t♦s ❢♦r❛♠ ♠✉✐t♦ ✉s❛❞♦s ♥♦ ❛♣r❡♥❞✐③❛❞♦ ❞❛ ▼❛t❡♠át✐❝❛ ♣♦r ♠❛✐s ❞❡ ❞♦✐s ♠✐❧ê♥✐♦s ✧✳ ❖ ú❧t✐♠♦ ❧✐✈r♦ ✭▲✐✈r♦ ❳■■■✮ é t♦t❛❧♠❡♥t❡ ❞❡❞✐❝❛❞♦ ❛♦s ❝✐♥❝♦ só❧✐❞♦s r❡❣✉❧❛r❡s✱ ❢❛t♦ q✉❡ ❧❡✈♦✉ ❛❧❣✉♥s ❤✐st♦r✐❛❞♦r❡s ❛ ❞✐③❡r q✉❡ ❖s ❊❧❡♠❡♥t♦s ❢♦r❛♠ ❝♦♠♣♦st♦s ❝♦♠♦ ✉♠❛ ❣❧♦r✐✜❝❛çã♦ ❞❛s ✜❣✉r❛s ❝ós♠✐❝❛s ♦✉ ♣❧❛tô♥✐❝❛s✳ ❖✉tr♦ ❝♦♠ ❡st✉❞♦s ♥♦ ❝❛♠♣♦ ❞❛ ●❡♦♠❡tr✐❛ ❊s♣❛❝✐❛❧ ❢♦✐ P❧❛tã♦✱ ❡♠❜♦r❛ ♥ã♦ t❡♥❤❛ ❞❛❞♦ ❝♦♥tr✐❜✉✐çã♦ ❞✐❣♥❛ ❞❡ ♥♦t❛ ❛ r❡s✉❧t❛❞♦s ♠❛t❡♠át✐❝♦s té❝♥✐❝♦s✱ ♣♦ré♠ s❡✉ ❡♥t✉s✐❛s♠♦ ♣❡❧♦ ❛ss✉♥t♦ ♦ ❢❡③ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ✧❝r✐❛❞♦r ❞❡ ♠❛t❡♠át✐❝♦s✧✳ ❊ss❡ ❢❛s❝í♥✐♦ t♦❞♦ ♣❡❧❛ ♠❛t❡♠át✐❝❛ ❝❡rt❛♠❡♥t❡ ✈❡✐♦ ❛ ♣❛rt✐r ❞❡ ✉♠❛ ✈✐s✐t❛ ❛ ✉♠ ❛♠✐❣♦ ♥❛ ❙✐❝í❧✐❛✱ ❆rq✉✐t❛s✳ ❚❛❧✈❡③ à ✈❡♥❡r❛çã♦ ❞♦s ♣✐t❛❣ór✐❝♦s ♣❡❧♦ ❞♦❞❡❝❛❡❞r♦ t❡♥❤❛ s✐❞♦ ♦ q✉❡ ❧❡✈♦✉ P❧❛tã♦ ❛ ❝♦♥s✐❞❡rá✲❧♦ ❝♦♠♦ ✉♠ sí♠❜♦❧♦ ❞♦ ✉♥✐✈❡rs♦✳ ❖s ❝✐♥❝♦ ♣♦❧✐❡❞r♦s r❡❣✉❧❛r❡s ❢♦r❛♠ ❝♦♥st❛♥t❡♠❡♥t❡ ❝❤❛♠❛❞♦s ❞❡ ✧❝♦r♣♦s ❝ós♠✐❝♦s✧ ♦✉ ✧só❧✐❞♦s ♣❧❛tô✲ ♥✐❝♦s✧ ❞❡✈✐❞♦ ❛ ♠❛♥❡✐r❛ ♣❡❧❛ q✉❛❧ P❧❛tã♦ ♦s ❛♣❧✐❝♦✉ ♣❛r❛ ❡①♣❧✐❝❛çã♦ ❞❡ ❢❡♥ô♠❡♥♦s ❝✐❡♥tí✜❝♦s✱ t❛❧✈❡③ t❛♠❜é♠ s❡❥❛ r❡s♣♦♥sá✈❡❧ ♣♦r ❛❧❣✉♥s ❝á❧❝✉❧♦s ❡♥❝♦♥tr❛❞♦s s♦❜r❡ ♦s ♣♦❧✐❡❞r♦s r❡❣✉❧❛r❡s ♥♦ ❧✐✈r♦ ❖s ❊❧❡♠❡♥t♦s✳ ✭✈❡❥❛ ❬✺❪ ♣á❣✐♥❛s ✻✷✲✻✸✮✳ ❚✐✈❡♠♦s ♦✉tr♦s ♠❛t❡♠át✐❝♦s ❣r❡❣♦s✱ ❡ ❝❧❛r♦ ❞❡ ♦✉tr♦s ♣♦✈♦s✱ q✉❡ ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ●❡♦♠❡tr✐❛ ❊s♣❛❝✐❛❧✱ ♣♦ré♠ ♦s ❣r❡❣♦s ❛❜s♦❧✈✐❛♠ ♠✉✐t♦s ❡❧❡♠❡♥t♦s ❞❡ ♦✉tr❛s ❝✉❧t✉r❛s✱ ❞❡ ♦✉tr❛ ❢♦r♠❛ ♥ã♦ t❡r✐❛♠ ❛♣r❡♥❞✐❞♦ tã♦ ❞❡♣r❡ss❛ ❞❡ ♠♦❞♦ ❛ t❡r ♣❛ss❛❞♦ à ❢r❡♥t❡ ❞❡ s❡✉s ♣r❡❞❡❝❡ss♦r❡s✱ ♠❛s ❛ t✉❞♦ q✉❡ t♦❝❛✈❛♠ ❞❛✈❛♠ ♠❛✐s ✈✐❞❛✳ ✭✈❡❥❛ ❬✺❪ ♣á❣✐♥❛ ✸✹✮✳ ✸

(17) ❋✐❣✉r❛ ✶✳✸✿ ❙ó❧✐❞♦s P❧❛tô♥✐❝♦s ❡ s❡✉s r❡s♣❡❝t✐✈♦s ❡❧❡♠❡♥t♦s ❞❛ ♥❛t✉r❡③❛✳ ❉❡♥tr❡ t♦❞♦s ♦s r❛♠♦s ❞❛ ♠❛t❡♠át✐❝❛ ❛ ●❡♦♠❡tr✐❛ é ❛ q✉❡ ♠❛✐s s♦❢r❡ ♠✉❞❛♥ç❛s ❞❡ ❣♦st♦ ❞❡ ✉♠❛ é♣♦❝❛ ♣❛r❛ ♦✉tr❛✳ ❙❡✉ ❛✉❣❡ ❢♦✐ ♥❛ ●ré❝✐❛ ❝❧áss✐❝❛ ❡ ❝❛✐✉ ❥✉♥t♦ ❝♦♠ ❘♦♠❛✱ t❡✈❡ ✉♠❛ r❡❝✉♣❡r❛çã♦ ♥❛ ➪r❛❜✐❛ ❡ ♥❛ ❊✉r♦♣❛ ❞❛ ❘❡♥❛s❝❡♥ç❛✱ ♥♦ sé❝✉❧♦ ❞❡③❡ss❡t❡ ❡st❡✈❡ ♥♦ ❧✐♠✐❛r ❞❡ ✉♠❛ ♥♦✈❛ ❡r❛✱ ♠❛s ♥♦✈❛♠❡♥t❡ ❢♦✐ ❡sq✉❡❝✐❞❛✱ ❛♦ ♠❡♥♦s ♣❡❧♦s ♣❡sq✉✐s❛❞♦r❡s ♠❛t❡♠át✐❝♦s✱ ♣♦r q✉❛s❡ ♠❛✐s ❞♦✐s sé❝✉❧♦s✱ ❛tr❛✈és ❞❡ ❡s❢♦rç♦s ❞❡ ▼♦♥❣❡ ❡ ❈❛r♥♦t ❤♦✉✈❡ ✉♠ r❡❛✈✐✈❛♠❡♥t♦ ❞❛ ❣❡♦♠❡tr✐❛ ♣✉r❛ ♥♦ ♣❡rí♦❞♦ ❞❛ r❡✈♦❧✉çã♦ ❋r❛♥❝❡s❛✱ ♣♦ré♠ ♥♦ sé❝✉❧♦ ❞❡③❡♥♦✈❡ ❡❧❛ t❡✈❡ ✉♠❛ r❡❞❡s❝♦❜❡rt❛ ❝♦♠♦ r❛♠♦ ✈✐✈♦ ❞❛ ♠❛t❡♠át✐❝❛✳ ✭✈❡❥❛ ❬✺❪ ♣á❣✐♥❛ ✸✽✼✮✳ ❍♦❥❡✱ ❝♦♠♦ ♥♦ ♣❛ss❛❞♦✱ ❛ ●❡♦♠❡tr✐❛ ❝♦♥t✐♥✉❛ ❛ s❡r ✉♠❛ ❢♦♥t❡ ❞❡ ❡♥r✐q✉❡❝✐♠❡♥t♦ ❞❡ r❛❝✐♦❝í♥✐♦ ❡ ❞♦s ❤á❜✐t♦s ❞❡ ♣❡♥s❛r✱ q✉❡ ♣❡r♠✐t❡♠ ❥✉st✐✜❝❛r ❛s ♥♦ss❛s ❛✜r♠❛çõ❡s ✭✈❡❥❛ ❬✶✹❪ ♣á❣✐♥❛ ✸✵✻✮ s✉❛s ❛♣❧✐❝❛çõ❡s sã♦ ✐♥ú♠❡r❛s✳ ❖s só❧✐❞♦s ❣❡♦♠étr✐❝♦s sã♦ ♦❜s❡r✈❛❞♦s✱ ♣♦r ❡①❡♠♣❧♦✱ ❡♠ ❞✐✈❡rs♦s ❧✉❣❛r❡s s❡❥❛ ❡♠ ❝♦♥str✉çõ❡s ♦✉ ❡♠ ♦❜❥❡t♦s ❞❡ ✉s♦ ❞✐ár✐♦✳ ❋✐❣✉r❛ ✶✳✹✿ P✐râ♠✐❞❡s ❞♦ ❊❣✐t♦ ❡ ❛ P✐râ♠✐❞❡ ❞♦ ♠✉s❡✉ ❞♦ ▲♦✉✈r❡✱ ❢♦r♠❛s ♣♦❧✐é❞r✐❝❛s✳ ✹

(18) ❈❛♣ít✉❧♦ ✷ ❖s ♣♦❧✐❡❞r♦s ♥♦s ❧✐✈r♦s ❞✐❞át✐❝♦s ❱ár✐❛s sã♦ ❛s ❞❡✜♥✐çõ❡s ♣❛r❛ ♦s ♣♦❧✐❡❞r♦s ❞✐s♣♦st❛s ♥♦s ❧✐✈r♦s ❞✐❞át✐❝♦s✳ ❙❡❣✉♥❞♦ ❇✐❛♥❝❤✐♥✐ ❡ P❛❝❝♦❧❛ ❬✹❪✱ ♣á❣✐♥❛ ✷✵✺✱ ❞❡✜♥✐♠♦s ♣♦❧✐❡❞r♦ ❝♦♠♦ à r❡❣✐ã♦ ❞♦ ❡s♣❛ç♦ ❧✐♠✐t❛❞❛ ♣♦r ♣♦❧í❣♦♥♦s ♣❧❛♥♦s✱ ❡ t❛✐s q✉❡ ❝❛❞❛ ✉♠❛ ❞❛s ❛r❡st❛s ❞❡ss❡s ♣♦❧í❣♦♥♦s ♣❡rt❡♥ç❛ ❛ ❞♦✐s ❡ s♦♠❡♥t❡ ❞♦✐s ❞❡❧❡s✳ ❊♠ ❇❛rr♦s♦ ❬✸❪✱ ♣á❣✐♥❛ ✶✻✹✱ é ❝❤❛♠❛❞♦ ❞❡ ♣♦❧✐❡❞r♦ ♦ só❧✐❞♦ ❢♦r♠❛❞♦ ♣❡❧❛ r❡✉♥✐ã♦ ❞❡ ✉♠❛ s✉♣❡r❢í❝✐❡ ❢❡❝❤❛❞❛ ❝♦♠ t♦❞♦s ♦s ♣♦♥t♦s ❞♦ ❡s♣❛ç♦ ❞❡❧✐♠✐t❛❞♦s ♣♦r ❡❧❛✳ ❊♠ ❉❛♥t❡ ❬✻❪✱ ♣á❣✐♥❛ ✸✻✵✱ ❞✐③ q✉❡ ♦ ♣♦❧✐❡❞r♦ é ❢♦r♠❛❞♦ ♣❡❧❛ r❡✉♥✐ã♦ ❞❡ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ r❡❣✐õ❡s ♣♦❧✐❣♦♥❛✐s ♣❧❛♥❛s ❝❤❛♠❛❞❛s ❢❛❝❡s ❡ ❛ r❡❣✐ã♦ ❞♦ ❡s♣❛ç♦ ❧✐♠✐t❛❞❛ ♣♦r ❡❧❛✱ ❛ss✐♠ ❝♦♠♦ ❘✐❜❡✐r♦ ❬✶✺❪✱ ♣á❣✐♥❛ ✻✽✱ ♣♦❧✐❡❞r♦s sã♦ só❧✐❞♦s ❧✐♠✐t❛❞♦s ♣♦r s✉♣❡r❢í❝✐❡s ♣❧❛♥❛s ♣♦❧✐❣♦♥❛✐s✳ P❛r❛ ■❡③③✐ ❬✾❪✱ ♣á❣✐♥❛ ✹✺✼✱ t❡♠♦s ♣♦❧✐❡❞r♦ ❝♦♠♦ só❧✐❞♦s ❧✐♠✐t❛❞♦s ♣♦r ♣♦rçõ❡s ❞❡ ♣❧❛♥♦s ✲ ♣♦❧í❣♦♥♦s ♣❧❛♥♦s ✲ ❞❡♥♦♠✐♥❛❞♦s ❢❛❝❡s✳ ❆s ❞❡✜♥✐çõ❡s ♥❡♠ s❡♠♣r❡ sã♦ s✉✜❝✐❡♥t❡s ♣❛r❛ q✉❡ ❛ ♠❛✐♦r✐❛ ❞♦s ❛❧✉♥♦ t❡♥❤❛♠ ✉♠❛ ✐❞❡✐❛ s♦❜r❡ ♦ q✉❡ é ✉♠ ♣♦❧✐❡❞r♦✱ ❛ ❛♣r❡s❡♥t❛çã♦ ❞❡ ✐❧✉str❛çõ❡s ❢❛❝✐❧✐t❛♠ s✉❛ ❝♦♠♣r❡❡♥sã♦✱ ♠❛s ❛✐♥❞❛✱ é ♥❡❝❡ssár✐♦ ✉♠ ❡st✉❞♦ ❞❛s ❢ór♠✉❧❛s q✉❡ ♣r♦❞✉③❡♠ ár❡❛s ❞❡ s✉♣❡r❢í❝✐❡s ❡ ✈♦❧✉♠❡s ❞♦s ♣♦❧✐❡❞r♦s✱ ❜❡♠ ❝♦♠♦ ♦ ❝❧áss✐❝♦ t❡♦r❡♠❛ ❞❡ ❊✉❧❡r✱ q✉❡ r❛r❛♠❡♥t❡ é ❞❡♠♦♥str❛❞♦ ♥♦s ❧✐✈r♦s ❞✐❞át✐❝♦s ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✳ ❱❛♠♦s ✉t✐❧✐③❛r ❛s ❞❡✜♥✐çõ❡s ❞❡ ▲✐♠❛ ❬✶✷❪ ♣á❣✐♥❛ ✷✽✸✳ ❉❡✜♥✐çã♦ ✷✳✶ P♦❧✐❡❞r♦ é ✉♠❛ r❡✉♥✐ã♦ ❞❡ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ♣♦❧í❣♦♥♦s ♣❧❛♥♦s ❝❤❛♠❛❞♦s ❢❛❝❡s ♦♥❞❡✿ ❛✮ ❈❛❞❛ ❧❛❞♦ ❞❡ ✉♠ ❞❡ss❡s ♣♦❧í❣♦♥♦s é t❛♠❜é♠ ❧❛❞♦ ❞❡ ✉♠✱ ❡ ❛♣❡♥❛s ✉♠✱ ♦✉tr♦ ♣♦❧í❣♦♥♦✳ ❜✮ ❆ ✐♥t❡rs❡çã♦ ❞❡ ❞✉❛s ❢❛❝❡s q✉❛✐sq✉❡r✱ ♦✉ é ✉♠ ❧❛❞♦ ❝♦♠✉♠✱ ♦✉ é ✈ért✐❝❡ ♦✉ é ✈❛③✐❛✳ ❝✮ ➱ s❡♠♣r❡ ♣♦ssí✈❡❧ ✐r ❞❡ ✉♠ ♣♦♥t♦ ❞❡ ✉♠❛ ❢❛❝❡ ❛ ✉♠ ♣♦♥t♦ ❞❡ q✉❛❧q✉❡r ♦✉tr❛✱ s❡♠ ♣❛ss❛r ♣♦r ♥❡♥❤✉♠ ✈ért✐❝❡✱ ♦✉ s❡❥❛✱ ❝r✉③❛♥❞♦ ❛♣❡♥❛s ❛r❡st❛s✳ ✺

(19) ❖ q✉❡ ❝♦♥s✐❞❡r❛r❡♠♦s ❝♦♠♦ ♣♦❧✐❡❞r♦ é ❛ s✉♣❡r❢í❝✐❡ ♣♦❧✐é❞r✐❝❛ ❢♦r♠❛❞❛ ♣❡❧♦s ♣♦❧í❣♦✲ ♥♦s ♥ã♦ ❛ r❡❣✐ã♦ ✐♥t❡r♥❛ ❧✐♠✐t❛❞❛ ♣♦r ❡❧❡s✳ ❋✐❣✉r❛ ✷✳✶✿ ❋✐❣✉r❛s P♦❧✐é❞r✐❝❛s✳ ❉❡ ❢❛t♦ é ❢á❝✐❧ ❡♥❝♦♥tr❛r♠♦s ❢♦r♠❛s ♣♦❧✐é❞r✐❝❛s ♥♦ ♥♦ss♦ ❞✐❛ ❛ ❞✐❛✱ ❛ss✐♠ t♦r♥❛✲ s❡ ❢á❝✐❧ ♣❛r❛ q✉❡ ♦ ❛❧✉♥♦ ❛ss♦❝✐❡ ❛ ❞❡✜♥✐çã♦ ❞❡ ♣♦❧✐❡❞r♦✱ ♣♦ré♠ ❡①✐st❡♠ t❛♠❜é♠ ✜❣✉r❛s q✉❡ ❡stã♦ ♥♦ R3✱ ♠❛s ♥ã♦ s❡❣✉❡♠ ❛❧❣✉♠ ♦✉ ❛❧❣✉♥s ❞♦s ✐t❡♥s ❞❛ ❉❡✜♥✐çã♦ ✷✳✶✱ ❝❤❛♠❛r❡♠♦s ❞❡ ✜❣✉r❛s ♥ã♦ ♣♦❧✐é❞r✐❝❛s✳ ❋✐❣✉r❛ ✷✳✷✿ ❋✐❣✉r❛s ♥ã♦ P♦❧✐é❞r✐❝❛s✳ ❆ss✐♠ t♦❞♦ ♣♦❧✐❡❞r♦ ♥♦ s❡♥t✐❞♦ ❞❛ ❉❡✜♥✐çã♦ ✷✳✶ ❧✐♠✐t❛ ✉♠❛ r❡❣✐ã♦ ❞♦ ❡s♣❛ç♦ ❝❤❛♠❛❞❛ ✐♥t❡r✐♦r ❞❡ss❡ ♣♦❧✐❡❞r♦✱ ❡ss❛s ✜❣✉r❛s ❣❡♦♠étr✐❝❛s ❡stã♦ ♥♦ R3✳ ◆❛ ❡❞✉❝❛çã♦ ❜ás✐❝❛ ♦s ♣♦❧✐❡❞r♦s tê♠ ♠❛✐♦r ê♥❢❛s❡ ♥♦ s❡❣✉♥❞♦ ❛♥♦ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ♣♦ré♠ ❛❧❣✉♥s ❛✉t♦r❡s ❥á ✐♥tr♦❞✉③❡♠ ♦ ❛ss✉♥t♦ ♥♦ ♣r✐♠❡✐r♦ ❛♥♦ ❝♦♠ ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ❝♦♠♦ ❡❧❡♠❡♥t♦s ❡ ❝❧❛ss✐✜❝❛çõ❡s ❞♦s ♣♦❧✐❡❞r♦s✳ ❙❡❣✉♥❞♦ P❛❧❤❛✲ r❡s ❬✶✹❪ ♣á❣✐♥❛ ✸✵✻✱ é ✉r❣❡♥t❡ ❡ ❢✉♥❞❛♠❡♥t❛❧ q✉❡ ❛ ●❡♦♠❡tr✐❛ r❡✢✐t❛✱ ♣♦r✈❡♥t✉r❛✱ ❤♦❥❡ ♠❛✐s ❞♦ q✉❡ ♥♦ ♣❛ss❛❞♦ r❡❝❡♥t❡✱ ❛s ♣r❡♦❝✉♣❛çõ❡s ❡❞✉❝❛❝✐♦♥❛✐s ❞❡ r❡❧❡✈â♥❝✐❛ ❡ r❡❛❧✐s♠♦✱ ♥♦♠❡❛❞❛♠❡♥t❡ ❛tr❛✈és ❞❡✿ ✻

(20) ✷✳✶✳ P❖▲■❊❉❘❖❙ ❈❖◆❱❊❳❖❙ ❼ ❱❡r❞❛❞❡✐r♦s ♣r♦❜❧❡♠❛s ❞♦ ❞✐❛✲❛✲❞✐❛ q✉❡ ❡♥✈♦❧✈❛♠ ✐❞❡✐❛s ❣❡♦♠étr✐❝❛s✱ ❡♠ ✈❡③ ❞❡ ❛♣❧✐❝❛çõ❡s ❛rt✐✜❝✐❛s❀ ❼ ❊①♣❧♦r❛çã♦ ❞❡ ❢♦r♠❛s ❞❡ r❡♣r❡s❡♥t❛çã♦ ❞♦ ♠❡✐♦ ❛♠❜✐❡♥t❡✱ ♣♦r ♠✉✐t♦ ❝♦♠♣❧✐✲ ❝❛❞♦ q✉❡ ✐ss♦ ♣❛r❡ç❛✳ ❖ ✉s♦ ❞❡ ♣❧❛♥t❛s✱ ❞❡ ♠❛♣❛s ♦✉ ❞❡ ❢♦t♦❣r❛✜❛s ♣❛r❡❝❡✲♥♦s ❛♣r♦♣r✐❛❞♦✱ ✉♠❛ ✈❡③ q✉❡ ♥♦ ♠✉♥❞♦ ♦♥❞❡ ❛ ❝r✐❛♥ç❛ s❡ ♠♦✈✐♠❡♥t❛ ❛ ●❡♦♠❡tr✐❛ é✱ ❡♠ ♣r✐♠❡✐r♦ ❧✉❣❛r✱ ❊s♣❛❝✐❛❧✱ ❛♥t❡s ❞❡ s❡r ♣❧❛♥❛❀ ❼ ❚r❛❜❛❧❤♦s ❣❡♦♠étr✐❝♦s ❝♦♠ ♦ r❡❝✉rs♦ às ♥♦✈❛s t❡❝♥♦❧♦❣✐❛s✳ ✷✳✶ P♦❧✐❡❞r♦s ❝♦♥✈❡①♦s ◆♦ í♥✐❝✐♦ ❞❡ss❡ ❝❛♣ít✉❧♦ r❡✈✐s❛♠♦s ❛s ✈ár✐❛s ❡ ❡q✉✐✈❛❧❡♥t❡s ♠❛♥❡✐r❛s ❝♦♠♦ ♦s ♣♦❧✐✲ ❡❞r♦s sã♦ ❞❡✜♥✐❞♦s ♥♦s ❧✐✈r♦s ❞✐❞át✐❝♦s ✉s❛❞♦s ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✳ ❆ s❡❣✉✐r✱ tr❛t❛♠♦s ❞♦s ❝❤❛♠❛❞♦s ♣♦❧✐❡❞r♦s ❝♦♥✈❡①♦s✳ ❈♦♥❢♦r♠❡ ❏❛♥♦s ❬✶✵❪✱ ♣á❣✐♥❛ ✽✹✱ ✉♠ ♣♦❧✐❡❞r♦ ❝♦♥✈❡①♦ ❡♥❝♦♥tr❛✲s❡ ✐♥t❡✐r❛♠❡♥t❡ ❞❡ ✉♠ ❧❛❞♦ ❞♦ ♣❧❛♥♦ q✉❡ ❝♦♥té♠ q✉❛❧q✉❡r ✉♠❛ ❞❡ s✉❛s ❢❛❝❡s✳ ◆❛ ✜❣✉r❛ ❛❜❛✐①♦ ♣♦❞❡♠♦s ♦❜s❡r✈❛r q✉❡ ♦ ♣♦❧✐❡❞r♦ ♥ã♦ ❝♦♥✈❡①♦ ❛♣r❡s❡♥t❛ ✉♠ ♣❧❛♥♦ q✉❡ ❝♦♥té♠ ✉♠❛ ❞❡ s✉❛s ❢❛❝❡s ❡ ❝♦rt❛ ♦✉tr❛✳ ❋✐❣✉r❛ ✷✳✸✿ P♦❧✐❡❞r♦s✳ P❛r❛ ❘✐❜❡✐r♦ ❬✶✺❪✱ ♣á❣✐♥❛ ✼✵✱ ✉♠ ♣♦❧✐❡❞r♦ é ❞✐t♦ ❝♦♥✈❡①♦ q✉❛♥❞♦ ✉♠ s❡❣♠❡♥t♦ ❞❡ r❡t❛ q✉❡ ❧✐❣❛ q✉❛✐sq✉❡r ❞♦✐s ❞❡ s❡✉s ♣♦♥t♦s ❡stá ✐♥t❡✐r❛♠❡♥t❡ ❝♦♥t✐❞♦ ♥❡❧❡✳ ❆❧é♠ ❞✐ss♦✱ ✉♠ ♣♦❧✐❡❞r♦ é ❝♦♥✈❡①♦ s❡ t♦❞❛ r❡t❛ ♥ã♦ ♣❛r❛❧❡❧❛ ❛ ♥❡♥❤✉♠❛ ❞❛s ❢❛❝❡s ❝♦rt❛ s✉❛s ❢❛❝❡s ❡♠ ❞♦✐s ♣♦♥t♦s ♥♦ ♠á①✐♠♦✳ ❆ ✜❣✉r❛ ❛❜❛✐①♦ ❛♣r❡s❡♥t❛ ❡①❡♠♣❧♦s ❞❡ ♣♦❧✐❡❞r♦s ❝♦♥✈❡①♦ ❡ ♥ã♦ ❝♦♥✈❡①♦✳ ❙❡❣✉♥❞♦ ▲✐♠❛ ❬✶✶❪✱ ♣á❣✐♥❛ ✷✽✹✱ ✉♠ ❝♦♥❥✉♥t♦ ❈ ❝♦♥t✐❞♦ ♥♦ ♣❧❛♥♦ ♦✉ ♥♦ ❡s♣❛ç♦✱ ❞✐③✲s❡ ❝♦♥✈❡①♦✱ q✉❛♥❞♦ q✉❛❧q✉❡r s❡❣♠❡♥t♦ ❞❡ r❡t❛ q✉❡ ❧✐❣❛ ❞♦✐s ♣♦♥t♦s ❞❡ ❈ ❡stá ✐♥t❡✐r❛♠❡♥t❡ ❝♦♥t✐❞♦ ❡♠ ❈✳ ❆ss✐♠ ♣♦❞❡♠♦s ✐♠❛❣✐♥❛r ❡ss❡ ❝♦♥❥✉♥t♦ ❈ t❛♥t♦ ♥♦ ❡s♣❛ç♦ tr✐❞✐♠❡♥s✐♦♥❛❧ ♦✉ ❜✐❞✐♠❡♥s✐♦♥❛❧✱ ❝♦♠♦ ♥♦ss❛ ❢♦♥t❡ ❞❡ ❡st✉❞♦ é ♥♦ R3 ♣♦❞❡♠♦s ✉s❛r ❛ s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦ t❛♠❜é♠ ♣r♦♣♦st❛ ♣♦r ▲✐♠❛ ❬✶✶❪✱ ♣á❣✐♥❛ ✷✽✹✳ ✼

(21) ✷✳✷✳ ❊▲❊▼❊◆❚❖❙ ❉❖❙ P❖▲■❊❉❘❖❙ ❋✐❣✉r❛ ✷✳✹✿ P♦❧✐❡❞r♦s ♥ã♦ ❝♦♥✈❡①♦ ✭à ❡sq✉❡r❞❛✮ ❡ P♦❧✐❡❞r♦ ❝♦♥✈❡①♦ ✭à ❞✐r❡✐t❛✮✳ ❉❡✜♥✐çã♦ ✷✳✷ ❯♠ ♣♦❧✐❡❞r♦ é ❞✐t♦ ❝♦♥✈❡①♦ s❡ q✉❛❧q✉❡r r❡t❛ ✭♥ã♦ ♣❛r❛❧❡❧❛ ❛ ♥❡♥❤✉♠❛ ❞❡ s✉❛s ❢❛❝❡s✮ ♦ ❝♦rt❛ ❡♠✱ ♥♦ ♠á①✐♠♦✱ ❞♦✐s ♣♦♥t♦s✳ ✷✳✷ ❊❧❡♠❡♥t♦s ❞♦s ♣♦❧✐❡❞r♦s ◆❡st❛ s❡çã♦ ✈❡r❡♠♦s ♦s ❡❧❡♠❡♥t♦s q✉❡ ❝♦♠♣õ❡♠ ♦s ♣♦❧✐❡❞r♦s✳ ❼ ❋❛❝❡s ❖s ♣♦❧í❣♦♥♦s ✷✱ ✸✱ ✹ ❡ ✺ ❞❛ ✜❣✉r❛ ❛❜❛✐①♦ sã♦ tr✐â♥❣✉❧♦s✱ ❡♥q✉❛♥t♦ ♦ ♣♦❧í❣♦♥♦ ✶ é ✉♠ q✉❛❞r❛❞♦✱ ♣♦❞❡♠♦s ✉♥í✲❧♦s ❢♦r♠❛♥❞♦ ✉♠ ♣♦❧✐❡❞r♦ P ✱ ❛♦ ❢♦r♠❛r ♦ ♣♦❧✐❡❞r♦ P ❡ss❡s ♣♦❧í❣♦♥♦s ❧✐♠✐t❛rã♦ ✉♠❛ r❡❣✐ã♦ ❞♦ ❡s♣❛ç♦✱ ❛ss✐♠ ♣♦❞❡♠♦s ❞❡✜♥✐r ❢❛❝❡s ❞❡ ✉♠ ♣♦❧✐❡❞r♦✳ ❋✐❣✉r❛ ✷✳✺✿ P♦❧í❣♦♥♦s ♥♦ ♣❧❛♥♦✳ ❉❡✜♥✐çã♦ ✷✳✸ ❆s ❢❛❝❡s ❞❡ ✉♠ ♣♦❧✐❡❞r♦ sã♦ ♦s ♣♦❧í❣♦♥♦s q✉❡ ❧✐♠✐t❛♠ ♦ ♣♦❧✐❡✲ ❞r♦✳ ✽

(22) ✷✳✷✳ ❊▲❊▼❊◆❚❖❙ ❉❖❙ P❖▲■❊❉❘❖❙ ❈♦♥❢♦r♠❡ ❛ ❉❡✜♥✐çã♦ ✷✳✶ ♦s ♣♦❧✐❡❞r♦s tê♠ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ❢❛❝❡s✳ ❼ ❆r❡st❛s ❖s ❧❛❞♦s ❞♦ ♣♦❧í❣♦♥♦ ✶ A1B1✱ B1C1✱ C1D1 ❡ D1A1 s❡rã♦ ❝♦♠✉♠ ❝♦♠ ♦s ❧❛❞♦s ❞♦s ♣♦❧í❣♦♥♦s ✷✱ ✸✱ ✹ ❡ ✺ A3B3✱ B2C2✱ C5D5 ❡ D4A4 r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❊ss❡s ❧❛❞♦s ❝♦♠✉♥s ❢♦r♠❛rã♦ ❛s ❛r❡st❛s AB✱ BC✱ CD ❡ DA ❞♦ ♣♦❧✐❡❞r♦ P ✳ ❋✐❣✉r❛ ✷✳✻✿ P♦❧✐❡❞r♦ P ♣❧❛♥✐✜❝❛❞♦✳ ❉❡✜♥✐çã♦ ✷✳✹ ❆r❡st❛ é ♦ ♥♦♠❡ ❞❛❞♦ ❛ ❝❛❞❛ ❧❛❞♦ ❞❛ ❢❛❝❡ ❞♦ ♣♦❧✐❡❞r♦✱ ❛ q✉❛❧ é ❝♦♠✉♠ ❛ s♦♠❡♥t❡ ❞✉❛s ❢❛❝❡s✳ ❼ ❱ért✐❝❡s P♦❞❡♠♦s ♦❜s❡r✈❛r q✉❡ ♥❛ ✜❣✉r❛ ❛❝✐♠❛ ♦s ✈ért✐❝❡s ❞♦ ♣♦❧í❣♦♥♦ ✶ s❡rã♦ ❝♦♠✉♥s ❝♦♠ ♦s ✈ért✐❝❡s ❞♦s ♣♦❧í❣♦♥♦s ✷✱ ✸✱ ✹ ❡ ✺❀ ✭♣♦r ❡①❡♠♣❧♦ ♦ ✈ért✐❝❡ A1 ❡ B1 ❞♦ ❧❛❞♦ A1B1 ❞♦ ♣♦❧í❣♦♥♦ ✶ s❡rã♦ ❝♦♠✉♥s ❝♦♠ ♦s ✈ért✐❝❡s A3 ❡ B3✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❞♦ ❧❛❞♦ A3B3 ❞♦ ♣♦❧í❣♦♥♦ ✸✮✳ ❋✐❣✉r❛ ✷✳✼✿ ❱✐st❛ s✉♣❡r✐♦r ❞♦ ♣♦❧✐❡❞r♦ P ✳ ✾

(23) ✷✳✸✳ ❘❊▲❆➬Õ❊❙ ❊◆❚❘❊ ❖❙ ❊▲❊▼❊◆❚❖❙ ❉❖❙ P❖▲■❊❉❘❖❙ ❉❡✜♥✐çã♦ ✷✳✺ ❱ért✐❝❡ ❞❡ ✉♠ ♣♦❧✐❡❞r♦ é ❝❛❞❛ ✉♠ ❞♦s ♣♦♥t♦s ❞❡ ✐♥t❡rs❡çã♦ ❞❡ ✸ ♦✉ ♠❛✐s ❛r❡st❛s✳ ❖ ✈ért✐❝❡ ❞❡ ❝❛❞❛ ❢❛❝❡ t❛♠❜é♠ é ✈ért✐❝❡ ❞♦ ♣♦❧✐❡❞r♦ P ✳ ❆q✉✐✱ t❡♠♦s ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s ❡♥❝♦♥tr❛❞❛s ❡♠ ❧✐✈r♦s ❞✐❞át✐❝♦s ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ♣❛r❛ ♦s ❡❧❡♠❡♥t♦s q✉❡ ❝♦♠♣õ❡♠ ♦ ♣♦❧✐❡❞r♦✳ ❙❡❣✉♥❞♦ ❉❛♥t❡ ❬✻❪✱ ♣á❣✐♥❛ ✸✻✵✱ ❝❛❞❛ ♣♦❧✐❡❞r♦ é ❢♦r♠❛❞♦ ♣❡❧❛ r❡✉♥✐ã♦ ❞❡ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ r❡❣✐õ❡s ♣♦❧✐❣♦♥❛✐s ♣❧❛♥❛s ❝❤❛♠❛❞❛s ❢❛❝❡s ❡ ❛ r❡❣✐ã♦ ❞♦ ❧✐♠✐t❛❞❛s ♣♦r ❡❧❛s✳ ❈❛❞❛ ❧❛❞♦ ❞❡ ✉♠❛ ❞❡ss❛s r❡❣✐õ❡s ♣♦❧✐❣♦♥❛❧ é t❛♠❜é♠ ❧❛❞♦ ❞❡ ✉♠❛ ♦✉tr❛ ú♥✐❝❛ r❡❣✐ã♦ ♣♦❧✐❣♦♥❛❧✳ ❆ ✐♥t❡rs❡❝çã♦ ❞❡ ❞✉❛s ❢❛❝❡s q✉❛✐sq✉❡r ♦✉ é ✉♠❛ ❧❛❞♦ ❝♦♠✉♠✱ ♦✉ é ✉♠ ✈ért✐❝❡✱ ♦✉ é ✈❛③✐❛✳ ❈❛❞❛ ❧❛❞♦ ❞❡ ✉♠❛ r❡❣✐ã♦ ♣♦❧✐❣♦♥❛❧✱ ❝♦♠✉♠ ❛ ❡①❛t❛♠❡♥t❡ ❞✉❛s ❢❛❝❡s é ❝❤❛♠❛❞♦ ❛r❡st❛ ❞♦ ♣♦❧✐❡❞r♦✱ ❡ ❝❛❞❛ ✈ért✐❝❡ ❞❡ ✉♠❛ ❢❛❝❡ é ✉♠ ✈ért✐❝❡ ❞♦ ♣♦❧✐❡❞r♦✳ P❛r❛ ❘✐❜❡✐r♦ ❬✶✺❪✱ ♣á❣✐♥❛ ✻✽✱ ❢❛❝❡s sã♦ ♦s ♣♦❧í❣♦♥♦s q✉❡ ❧✐♠✐t❛♠ ♦s ♣♦❧✐❡❞r♦s✳ ❚♦❞♦ ♣♦❧✐❡❞r♦ t❡♠ ✉♠❛ q✉❛♥t✐❞❛❞❡ ✜♥✐t❛ ❞❡ ❢❛❝❡s✳ ❆r❡st❛ é ♦ ♥♦♠❡ q✉❡ s❡ ❞á ❛ ❝❛❞❛ ❧❛❞♦ ❞❡ ✉♠❛ ❢❛❝❡ ❞♦ ♣♦❧✐❡❞r♦✳ ❈❛❞❛ ❛r❡st❛ ❞❡ ✉♠ ♣♦❧✐❡❞r♦ é ❝♦♠✉♠ ❛ s♦♠❡♥t❡ ❞✉❛s ❢❛❝❡s✳ ❱ért✐❝❡ é ❝❛❞❛ ✉♠ ❞♦s ♣♦♥t♦s ❞❡ ✐♥t❡rs❡❝çã♦ ❞❡ ✸ ♦✉ ♠❛✐s ❛r❡st❛s✳ ❖ ✈ért✐❝❡ ❞❡ ❝❛❞❛ ❢❛❝❡ t❛♠❜é♠ é ♦ ✈ért✐❝❡ ❞♦ ♣♦❧✐❡❞r♦✳ ✷✳✸ ❘❡❧❛çõ❡s ❡♥tr❡ ♦s ❊❧❡♠❡♥t♦s ❞♦s P♦❧✐❡❞r♦s P♦❞❡♠♦s r❡❧❛❝✐♦♥❛r ♦s ❡❧❡♠❡♥t♦s ❞❡ ✉♠ ♣♦❧✐❡❞r♦ ❝♦♥✈❡①♦ q✉❛❧q✉❡r✳ ❆s r❡❧❛çõ❡s ❞❡s❝r✐t❛s ❢♦r❛♠ ❜❛s❡❛❞❛s ♥❛q✉❡❧❛s ❛♣r❡s❡♥t❛❞❛s ♣♦r ▲✐♠❛ ❬✶✷❪✱ ♣á❣✐♥❛s ✷✽✹ ✲ ✷✽✺✳ ◆❛ s❡çã♦ ❛♥t❡r✐♦r ❞❡✜♥✐♠♦s ♦s ❡❧❡♠❡♥t♦s q✉❡ ❝♦♠♣õ❡♠ ✉♠ ♣♦❧✐❡❞r♦✱ ✈❛♠♦s ❛q✉✐ r❡♣r❡s❡♥tá✲❧♦s ❝♦♠♦ A✱ ♦ ♥ú♠❡r♦ ❞❡ ❛r❡st❛s✱ V ♦ ♥ú♠❡r♦ ❞❡ ✈ért✐❝❡s ❡ ♣♦r F ♦ ♥ú♠❡r♦ ❞❡ ❢❛❝❡s ❞❡ ✉♠ ♣♦❧✐❡❞r♦ q✉❛❧q✉❡r✳ P♦❞❡♠♦s ❛✐♥❞❛ ✈❡r✐✜❝❛r q✉❡ ❛s ❢❛❝❡s ❞❡ ✉♠ ♣♦❧✐❡❞r♦ ♣♦❞❡♠ s❡r ❝♦♠♣♦st❛s ♣♦r ♣♦❧í❣♦♥♦s ❞✐❢❡r❡♥t❡s✱ r❡♣r❡s❡♥t❛r❡♠♦s ♣♦r Fn ♦ ♥ú♠❡r♦ ❞❡ ❢❛❝❡s q✉❡ ♣♦ss✉❡♠ n ❧❛❞♦s✱ ❝♦♠ n ≥ 3✱ ❛ss✐♠ t❡♠♦s F3 ❢❛❝❡ tr✐❛♥❣✉❧❛r✱ F4 ❢❛❝❡ q✉❛❞r❛♥❣✉❧❛r ❡ ❛ss✐♠ s✉❝❡ss✐✈❛♠❡♥t❡✳ ❙❡❣✉❡ ❞❛ ❉❡✜♥✐çã♦ ✷✳✶ q✉❡ ♦ ♥ú♠❡r♦ F ❞❡ ❢❛❝❡s é ✜♥✐t♦✱ ❡ ❛❧é♠ ❞✐ss♦ F = F3 + F4 + ... + Fn. ❉❛ ♠❡s♠❛ ❢♦r♠❛ ♣♦❞❡♠♦s ✈❡r✐✜❝❛r q✉❡ ♣❛r❛ ❝❛❞❛ ✈ért✐❝❡ V ❞❡ ✉♠ ♣♦❧✐❡❞r♦ ❝♦♥❝♦rr❡♠ p ❛r❡st❛s✱ ❝♦♠♦ ♣❛r❛ ❝❛❞❛ ✈ért✐❝❡ ♦ ♠❡♥♦r ♥ú♠❡r♦ ❞❡ ❛r❡st❛s q✉❡ ❝♦♥✲ ❝♦rr❡♠ ♣❛r❛ ❡❧❡ é três✱ t❡♠♦s q✉❡ p ≥ 3 ❡ Vp ♦ ♥ú♠❡r♦ ❞❡ ✈ért✐❝❡s q✉❡ ❝♦♥❝♦rr❡♠ p ❛r❡st❛s✱ ♣♦r ❡①❡♠♣❧♦ V3 é ♦ ♥ú♠❡r♦ ❞❡ ✈ért✐❝❡s ♥♦ q✉❛❧ ❝♦♥❝♦rr❡♠ três ❛r❡st❛s✱ V4 ♦ ♥ú♠❡r♦ ❞❡ ✈ért✐❝❡s ♥♦ q✉❛❧ ❝♦♥❝♦rr❡♠ q✉❛tr♦ ❛r❡st❛s ❡ ❛ss✐♠ s✉❝❡ss✐✈❛♠❡♥t❡✳ P♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ ✶✵

(24) ✷✳✸✳ ❘❊▲❆➬Õ❊❙ ❊◆❚❘❊ ❖❙ ❊▲❊▼❊◆❚❖❙ ❉❖❙ P❖▲■❊❉❘❖❙ V = V3 + V4 + ... + Vp. ✷✳✸✳✶ ❋❛❝❡s ❡ ❆r❡st❛s ❱❛♠♦s ✐♠❛❣✐♥❛r ✉♠ ♣♦❧✐❡❞r♦ ❝♦♠ t♦❞❛s ❛s s✉❛s ❢❛❝❡s s❡♣❛r❛❞❛s ❡ ❞✐s♣♦st❛s s♦✲ ❜r❡ ✉♠ ♣❧❛♥♦✳ ❙❡ q✉✐s❡r♠♦s s❛❜❡r q✉❛♥t♦s ❧❛❞♦s ♣♦ss✉❡♠ t♦❞♦s ♦s ♣♦❧í❣♦♥♦s q✉❡ ❢♦r♠❛ ✉♠ ♣♦❧✐❡❞r♦ ❜❛st❛ ♠✉❧t✐♣❧✐❝❛r♠♦s ❛s ❢❛❝❡s tr✐❛♥❣✉❧❛r❡s ✭tr✐â♥❣✉❧♦s✮ ♣♦r três✱ ❛s ❢❛❝❡s q✉❛❞r❛♥❣✉❧❛r❡s ✭q✉❛❞r✐❧át❡r♦s✮ ♣♦r q✉❛tr♦✱ ❡ ❛ss✐♠ s✉❝❡ss✐✈❛♠❡♥t❡ ❡ ❞❡♣♦✐s s♦♠❛r♠♦s t♦❞♦s ♦s r❡s✉❧t❛❞♦s✱ ♣♦ré♠ ❝❛❞❛ ❛r❡st❛ ❞♦ ♣♦❧✐❡❞r♦ é ❧❛❞♦ ❝♦♠✉♠ ❞❡ ❡①❛✲ t❛♠❡♥t❡ ❞✉❛s ❢❛❝❡s✱ ❛ss✐♠ ❛ s♦♠❛ ❞❡ t♦❞♦s ♦s ❧❛❞♦s ❞❛s ❢❛❝❡s r❡s✉❧t❛ ♦ ❞♦❜r♦ ❞❡ ❛r❡st❛s✳ ▲♦❣♦✱ 3F3 + 4F4 + 5F5 + · · · = 2A. nF P♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡✿ nF = 2A✱ ✈❛❧❡ ♣❛r❛ t♦❞♦ ♣♦❧✐❡❞r♦ r❡❣✉❧❛r✱ ♦♥❞❡ n é ♦ ♥ú♠❡r♦ ❞❡ ❧❛❞♦s ❞❡ ❝❛❞❛ ❢❛❝❡✳ ✷✳✸✳✷ ❱ért✐❝❡s ❡ ❆r❡st❛s P♦❞❡♠♦s t❛♠❜é♠ ❞❡t❡r♠✐♥❛r ♦ ♥ú♠❡r♦ ❞❡ ❛r❡st❛s ❝♦♠ ❛ ♦❜s❡r✈❛çã♦ ♥♦s ✈ért✐❝❡s ❞❡ ✉♠ ♣♦❧✐❡❞r♦✳ ❙❡ ❝♦♥t❛r♠♦s q✉❛♥t❛s ❛r❡st❛s ❝♦♥❝♦rr❡♠ ❡♠ ❝❛❞❛ ✈ért✐❝❡s ✈❛♠♦s ♦❜s❡r✈❛r q✉❡ ❝❛❞❛ ❛r❡st❛ s❡rá ❝♦♥t❛❞❛ ❞✉❛s ✈❡③❡s✱ ❡♠ ✉♠ ❡①tr❡♠♦ ❡ ♥♦ ♦✉tr♦✱ ❛ss✐♠ s❡ s♦♠❛r♠♦s ♦ r❡s✉❧t❛❞♦ ❞❛s ❛r❡st❛s q✉❡ ❝♦♥❝♦rr❡♠ ❡♠ ❝❛❞❛ ✈ért✐❝❡s t❡r❡♠♦s ♦ ❞♦❜r♦ ❞♦ ♥ú♠❡r♦ ❞❡ ❛r❡st❛s✳ ❆ss✐♠ t❡♠♦s✱ 3V3 + 4V4 + 5V5 + · · · = 2A. pV P♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡✿ pV = 2A✱ ✈❛❧❡ ♣❛r❛ t♦❞♦ ♣♦❧✐❡❞r♦ r❡❣✉❧❛r✱ ♦♥❞❡ p é ♦ ♥ú♠❡r♦ ❞❡ ❛r❡st❛s q✉❡ ❝♦♥❝♦rr❡♠ ❡♠ ❝❛❞❛ ✈ért✐❝❡ ❞❡ ✉♠ ♣♦❧✐❡❞r♦✳ ❙❡❣✉❡ ❞❛s r❡❧❛çõ❡s ❛❝✐♠❛ q✉❡ ♣♦❞❡♠♦s ❝♦♠♣❛rá✲❧❛s ❡ ❝♦♥❝❧✉ír♠♦s q✉❡✿ 2A = nF = pV. ✷✳✸✳✸ ❘❡❧❛çã♦ ❞❡ ❊✉❧❡r ◆❡st❛ s❡çã♦ s❡rá ❞❡♠♦♥str❛❞♦ ♦ t❡♦r❡♠❛ ❞❡ ❊✉❧❡r✱ q✉❡ ♣❡❧❛ s✉❛ ❛♣❧✐❝❛çã♦ s✐♠✲ ♣❧❡s✱ ❞❡✐①❛ ♦s ❛❧✉♥♦s ❝✉r✐♦s♦s ♣r✐♥❝✐♣❛❧♠❡♥t❡ q✉❛♥❞♦ ❡❧❡s t❡♠ ❡♠ ♠ã♦s ✉♠ ♣♦❧✐❡❞r♦ ❝♦♥✈❡①♦ q✉❛❧q✉❡r ❡ ❢❛③❡♠ ❛ ❝♦♥t❛❣❡♠ ❞♦s ❡❧❡♠❡♥t♦s✳ P❛r❛ t❛❧ ❞❡♠♦♥str❛çã♦ ❢♦✐ ✉t✐❧✐③❛❞❛ ❛ ❛♣r❡s❡♥t❛❞❛ ♣♦r ▲✐♠❛ ❬✶✷❪✱ ♣á❣✐♥❛s ✷✽✼✲✷✾✵✱ ✈❛❧❡ s❛❧✐❡♥t❛r q✉❡ ❡❧❛ s❡❣✉✐✉ q✉❛s❡ ✐♥t❡❣r❛❧♠❡♥t❡ ❛ ♣✉❜❧✐❝❛❞❛ ♣❡❧♦ ♣r♦❢❡ss♦r ❆③❛♠❜✉❥❛ ❋✐❧❤♦ ❬✷❪✳ ✶✶

(25) ✷✳✸✳ ❘❊▲❆➬Õ❊❙ ❊◆❚❘❊ ❖❙ ❊▲❊▼❊◆❚❖❙ ❉❖❙ P❖▲■❊❉❘❖❙ A V F❚❡♦r❡♠❛ ✷✳✶ ❉❛❞♦ ✉♠ ♣♦❧✐❡❞r♦ ❝♦♥✈❡①♦ ❝♦♠ ❛r❡st❛s✱ ✈ért✐❝❡s ❡ ❢❛❝❡s✱ t❡♠✲ s❡ V + F − A = 2. Pr♦✈❛✿ ■♥✐❝✐❛❧♠❡♥t❡ ✈❛♠♦s ❝❛❧❝✉❧❛r ❛ s♦♠❛ ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞❡ t♦❞❛s ❛s ❢❛❝❡s ❞❡ ✉♠ ♣♦❧✐❡❞r♦ ❝♦♥✈❡①♦ P ✱ ❡ss❛s ❢❛❝❡s s❡rã♦ ♥✉♠❡r❛❞❛s ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛ F1✱ F2✱ F3✱ ✳✳✳✱ Fk ❡ ✉♠ ♥ú♠❡r♦ n q✉❡ r❡♣r❡s❡♥t❛ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❧❛❞♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ❛s ❢❛❝❡s n1✱ n2✱ n3✱ ✳✳✳✱ nk✳ P❛r❛ ❝❛❧❝✉❧❛r ❛ s♦♠❛ ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞❡ ✉♠ ♣♦❧í❣♦♥♦ t❡♠♦s Si = 180 ◦ · (n − 2) ❢ór♠✉❧❛ ✈✐st❛ ♥♦ ❊♥s✐♥♦ ❋✉♥❞❛♠❡♥t❛❧✱ ♦✉ s❡❥❛✱ Si = π(n − 2)✳ ❆ s♦♠❛ ❞❡ t♦❞♦s ♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞♦ ♣♦❧✐❡❞r♦ P é ❞❛❞❛ ♣❡❧❛ ❡①♣r❡ssã♦ Si = π(n1 − 2) + π(n2 − 2) + ... + π(nk − 2) = π(n1 + n2 + ... + nk) − π(2 + 2 + ... + 2). ❈♦♠♦ ❝❛❞❛ ❛r❡st❛ ❞❡ ✉♠ ♣♦❧✐❡❞r♦ é ❝♦♠✉♠ ❛ ❛♣❡♥❛s ❞✉❛s ❢❛❝❡s✱ ❧♦❣♦ ❛ s♦♠❛ ❞♦s ni✬s ❧❛❞♦s ❞❡ ✉♠ ♣♦❧í❣♦♥♦ é ✐❣✉❛❧ ❛♦ ❞♦❜r♦ ❞♦ ♥ú♠❡r♦ ❞❡ ❛r❡st❛s✱ ♦✉ s❡❥❛ (n1 + n2 + ... + nk) = 2A✳ ❖ ♦✉tr♦ ♣❛rê♥t❡s❡ ❝♦♠♣♦st♦s ♣♦r s♦♠❛ ❞❡ ✷ ♣♦❞❡ s❡r r❡❡s❝r✐t♦ ❛ss✐♠ ✿ 2(1 + 1 + ...1) ♦♥❞❡ ❛ s♦♠❛ ❞❡ ♣❛r❝❡❧❛s ✉♥s é ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❢❛❝❡s q✉❡ ❝♦♠♣õ❡♠ ♦ ♣♦❧✐❡❞r♦✱ ❞❛í ❝♦♥❝❧✉í♠♦s q✉❡ (2 + 2 + ... + 2) = 2F ✳ ❊♥tã♦ t❡♠♦s Si = π2A − π2F = 2π(A − F ). ✭✷✳✶✮ P❛r❛ ♦ ♣ró①✐♠♦ ♣❛ss♦ ❞❛ ❞❡♠♦♥str❛çã♦ ✈❛♠♦s t♦♠❛r ✉♠ ♣❧❛♥♦ H ❤♦r✐③♦♥t❛❧ s♦❜ ♦ ♣♦❧✐❡❞r♦ P ❡ ✉♠❛ r❡t❛ r ♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ ♣❧❛♥♦ H t❛❧ q✉❡ ❛ r❡t❛ r ♥ã♦ s❡❥❛ ♣❛r❛❧❡❧❛ ❛ ♥❡♥❤✉♠❛ ❞❛s ❢❛❝❡s ❞♦ ♣♦❧✐❡❞r♦ P ✳ ❈♦❧♦❝❛♠♦s s♦❜r❡ ♦ ♣♦❧✐❡❞r♦ P ✉♠❛ ❢♦♥t❡ ❧✉♠✐♥♦s❛ ✭♣❛r❛ ♠❡❧❤♦r ❝♦♠♣r❡❡♥ssã♦ ❞♦ ❡①❡♠♣❧♦ ✐♠❛❣✐♥❡ ♦ s♦❧ ❛ ♣✐♥♦✱ ♦✉ s❡❥❛✱ ♦ ♣♦♥t♦ ♠❛✐s ❡❧❡✈❛❞♦ ❞♦ s♦❧✮ ❡ss❛ ❢♦♥t❡ ♣r♦❞✉③✐rá s♦❜r❡ ♦ ♣❧❛♥♦ H ✉♠❛ s♦♠❜r❛ P ′ ❞♦ ♣♦❧✐❡❞r♦ P ✳ ❊ss❛ s♦♠❜r❛ P ′ t❡♠ ❝♦♥t♦r♥♦ ❞❡ ✉♠ ♣♦❧í❣♦♥♦ ❝♦♥✈❡①♦ K′ s♦♠❜r❛ ❞❡ ✉♠ ♣♦❧✐❣♦♥❛❧ ❢❡❝❤❛❞❛ K✱ ❛ q✉❛❧ ❝❤❛♠❛r❡♠♦s ❞❡ ❝♦♥t♦r♥♦ ❛♣❛r❡♥t❡ ❞♦ ♣♦❧✐❡❞r♦ P ✱ ❢♦r♠❛❞❛ ♣♦r ❛r❡st❛s ❞❡ P ✳ ❈❛❞❛ ♣♦♥t♦ ❞♦ ♣♦❧í❣♦♥♦ K′ é ✉♠ ú♥✐❝♦ ♣♦♥t♦ ❞♦ ♣♦❧✐❡❞r♦ P ✱ ❥á ❛ s♦♠❜r❛ P ′ é s♦♠❜r❛ ❞❡ ❡①❛t❛♠❡♥t❡ ❞♦✐s ♣♦♥t♦s ❞❡ P ✳ ❯♠ ❞❡ss❡s ♣♦♥t♦s ✜❝❛ ♥❛ r❡❣✐ã♦ ✐❧✉♠✐♥❛❞❛ ❞❡ P ✱ r❡❣✐ã♦ q✉❡ ✜❝❛ ♠❛✐s ♣ró①✐♠❛ ❞❛ ❢♦♥t❡ ❧✉♠✐♥♦s❛ ❛ q✉❛❧ ❝❤❛♠❛r❡♠♦s ❞❡ P1 t♦❞♦ ♣♦♥t♦ ❞❡ss❛ r❡❣✐ã♦ é ❝❤❛♠❛❞♦ ❞❡ ♣♦♥t♦ ✐❧✉♠✐♥❛❞♦ ❡ ♦ ♦✉tr♦ ♣♦♥t♦ ✜❝❛rá ♥❛ r❡❣✐ã♦ ♠❡♥♦s ✐❧✉♠✐♥❛❞❛ ❞❡ P ✱ r❡❣✐ã♦ q✉❡ ✜❝❛ ♠❛✐s ♣ró①✐♠❛ ❞♦ ♣❧❛♥♦ H✱ ❛ q✉❛❧ ❝❤❛♠❛r❡♠♦s ❞❡ P2 t♦❞♦ ♣♦♥t♦ ❞❡ss❛ r❡❣✐ã♦ s❡rá ❝❤❛♠❛❞♦ ❞❡ ♣♦♥t♦ s♦♠❜r✐♦✳ ❉❡♣♦✐s ❞❡ss❛s ❝♦♥s✐❞❡r❛çõ❡s ✈❛♠♦s ❝❛❧❝✉❧❛r ♥♦✈❛♠❡♥t❡ ❛ s♦♠❛ ❞♦s â♥❣✉❧♦s ✐♥✲ t❡r♥♦s ❞❡ t♦❞❛s ❛s ❢❛❝❡s ❞❡ P ✳ ❉❡✈❡♠♦s ❧❡♠❜r❛r q✉❡ ❛ s♦♠❛ ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞❡ ✉♠❛ ❢❛❝❡ ❞❡ P é ✐❣✉❛❧ s♦♠❛ â♥❣✉❧♦s ✐♥t❡r♥♦s ❞❛ s♦♠❜r❛ ❞❡ss❛ ❢❛❝❡✳ ✶✷

(26) ✷✳✸✳ ❘❊▲❆➬Õ❊❙ ❊◆❚❘❊ ❖❙ ❊▲❊▼❊◆❚❖❙ ❉❖❙ P❖▲■❊❉❘❖❙ ❆ss✐♠ t❡♠♦s✱ V = V0 + V1 + V2 ♦♥❞❡ V0✱ V1✱ V2 sã♦ r❡s♣❡❝t✐✈❛♠❡♥t❡ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ✈ért✐❝❡s ❞♦ ❝♦♥t♦r♥♦ ❛♣❛r❡♥t❡✱ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ✈ért✐❝❡s ❞❡ P1 ❡ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ✈ért✐❝❡s ❞❡ P2✳ P❛r❛ ❝❛❧❝✉❧❛r ❛ s♦♠❛ ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞♦ ♣♦❧✐❡❞r♦ P t❡♠♦s Si = S1 + S2✱ ♦♥❞❡ S1 é ❛ s♦♠❛ ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞❛ r❡❣✐ã♦ ✐❧✉♠✐♥❛❞❛ ❡ S2 é ❛ s♦♠❛ ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞❛ r❡❣✐ã♦ s♦♠❜r✐❛✳ ❉❛❞❛s ♣♦r ❆♥❛❧♦❣❛♠❡♥t❡ t❡♠♦s S1 = 2πV1 + π(V0 − 2). ❊♥tã♦ S2 = 2πV2 + π(V0 − 2). Si = S1 + S2 = [2πV1 + π(V0 − 2)] + [2πV2 + π(V0 − 2)] = 2πV1 + 2πV2 + π(V0 − 2) + π(V0 − 2) = 2πV1 + 2πV2 + 2π(V0 − 2) = 2π(V1 + V2 + V0 − 2). ❖r❛✱ V1 + V2 + V0 = V ✳ ❉❛í t❡♠♦s Si = 2π(V − 2). ❯s❛♥❞♦ ❛s ❡q✉❛çõ❡s ✭✸✳✶✮ ❡ ✭✸✳✷✮✿ ✭✷✳✷✮ Si = Si 2π(A − F ) = 2π(V − 2) A−F = V −2 V − A + F = 2, ❞❡ss❛ ❢♦r♠❛ ❝♦♥❝❧✉í♠♦s ❛ ♣r♦✈❛✳ ✶✸

(27) ❈❛♣ít✉❧♦ ✸ ❆ ❘❡❧❛çã♦ ❞❡ ❊✉❧❡r ◆❡st❡ ❝❛♣ít✉❧♦ s❡rá ❞❡♠♦♥str❛❞♦ ♦ t❡♦r❡♠❛ ❞❡ ❊✉❧❡r✱ ♣❡❧❛ s✉❛ ❛♣❧✐❝❛çã♦ s✐♠✲ ♣❧❡s ❞❡✐①❛ ♦s ❛❧✉♥♦s ❝✉r✐♦s♦s ♣r✐♥❝✐♣❛❧♠❡♥t❡ q✉❛♥❞♦ ❡❧❡s t❡♠ ❡♠ ♠ã♦s ✉♠ ♣♦❧✐❡❞r♦ ❝♦♥✈❡①♦ q✉❛❧q✉❡r❡ ❢❛③❡♠ ❛ ❝♦♥t❛❣❡♠ ❞♦s ❡❧❡♠❡♥t♦s✳ P❛r❛ t❛❧ ❞❡♠♦♥str❛çã♦ ❢♦✐ ✉t✐✲ ❧✐③❛❞❛ ❛ ❛♣r❡s❡♥t❛❞❛ ♣♦r ▲✐♠❛ ❬✶✷❪ ♣á❣✐♥❛s ✷✽✼✲✷✾✵✱ ✈❛❧❡ s❛❧✐❡♥t❛r q✉❡ ❡❧❛ s❡❣✉✐✉ q✉❛s❡ ✐♥t❡❣r❛❧♠❡♥t❡ ❛ ♣✉❜❧✐❝❛❞❛ ♣❡❧♦ ♣r♦❢❡ss♦r ❆③❛♠❜✉❥❛ ❋✐❧❤♦ ❬✷❪✳ A V F❚❡♦r❡♠❛ ✸✳✶ ❉❛❞♦ ✉♠ ♣♦❧✐❡❞r♦ ❝♦♥✈❡①♦ ❝♦♠ ❛r❡st❛s✱ ✈ért✐❝❡s ❡ ❢❛❝❡s✱ t❡♠✲ s❡ V + F − A = 2. Pr♦✈❛✿ ■♥✐❝✐❛❧♠❡♥t❡ ✈❛♠♦s ❝❛❧❝✉❧❛r ❛ s♦♠❛ ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞❡ t♦❞❛s ❛s ❢❛❝❡s ❞❡ ✉♠ ♣♦❧✐❡❞r♦ ❝♦♥✈❡①♦ P ✱ ❡ss❛s ❢❛❝❡s s❡rã♦ ♥✉♠❡r❛❞❛s ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛ F1✱ F2✱ F3✱ ✳✳✳✱ Fk ❡ ✉♠ ♥ú♠❡r♦ n q✉❡ r❡♣r❡s❡♥t❛ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❧❛❞♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ❛s ❢❛❝❡s n1✱ n2✱ n3✱ ✳✳✳✱ nk✳ P❛r❛ ❝❛❧❝✉❧❛r ❛ s♦♠❛ ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞❡ ✉♠ ♣♦❧í❣♦♥♦ t❡♠♦s Si = 180 ◦ · (n − 2) ✈✐st❛ ♥♦ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧✱ ♦✉ s❡❥❛✱ Si = π(n − 2)✳ ❆ s♦♠❛ ❞❡ t♦❞♦s ♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞♦ ♣♦❧✐❡❞r♦ P é ❞❛❞❛ ♣❡❧❛ ❡①♣r❡ssã♦ Si = π(n1 − 2) + π(n2 − 2) + ... + π(nk − 2) = π(n1 + n2 + ... + nk) − π(2 + 2 + ... + 2). ❈♦♠♦ ❝❛❞❛ ❛r❡st❛ ❞❡ ✉♠ ♣♦❧✐❡❞r♦ é ❝♦♠✉♠ ❛ ❛♣❡♥❛s ❞✉❛s ❢❛❝❡s✱ ❧♦❣♦ ❛ s♦♠❛ ❞♦s n ✭❡♥❡s✮ ❧❛❞♦s ❞❡ ✉♠ ♣♦❧í❣♦♥♦ é ✐❣✉❛❧ ❛♦ ❞♦❜r♦ ❞♦ ♥ú♠❡r♦ ❞❡ ❛r❡st❛s✱ ♦✉ s❡❥❛ (n1 + n2 + ... + nk) = 2A✳ ❖ ♦✉tr♦ ♣❛rê♥t❡s❡ ❝♦♠♣♦st♦s ♣♦r ✷ ♣♦❞❡ s❡r r❡❡s❝r✐t♦ ❛ss✐♠ ✿ 2(1 + 1 + ...1) ♦♥❞❡ ❛ s♦♠❛ ❞❡ ♣❛r❝❡❧❛s ✉♥s é ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❢❛❝❡s q✉❡ ❝♦♠♣õ❡♠ ♦ ♣♦❧✐❡❞r♦✱ ❞❛í ❝♦♥❝❧✉í♠♦s q✉❡ (2 + 2 + ... + 2) = 2F ✳ ❊♥tã♦ t❡♠♦s Si = π2A − π2F = 2π(A − F ). ✶✹ ✭✸✳✶✮

(28) P❛r❛ ♦ ♣ró①✐♠♦ ♣❛ss♦ ❞❛ ❞❡♠♦♥str❛çã♦ ✈❛♠♦s t♦♠❛r ✉♠ ♣❧❛♥♦ H ❤♦r✐③♦♥t❛❧ s♦❜ ♦ ♣♦❧✐❡❞r♦ P ❡ ✉♠❛ r❡t❛ r ♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ ♣❧❛♥♦ H t❛❧ q✉❡ ❛ r❡t❛ r ♥ã♦ s❡❥❛ ♣❛r❛❧❡❧❛ ❛ ♥❡♥❤✉♠❛ ❞❛s ❢❛❝❡s ❞♦ ♣♦❧✐❡❞r♦ P ✳ ❈♦❧♦❝❛♠♦s s♦❜r❡ ♦ ♣♦❧✐❡❞r♦ P ✉♠❛ ❢♦♥t❡ ❧✉♠✐♥♦s❛ ✭♣❛r❛ ♠❡❧❤♦r ❝♦♠♣r❡❡♥ssã♦ ❞♦ ❡①❡♠♣❧♦ ✐♠❛❣✐♥❡ ♦ s♦❧ ❛ ♣✐♥♦✮ ❡ss❛ ❢♦♥t❡ ♣r♦❞✉③✐rá s♦❜r❡ ♦ ♣❧❛♥♦ H ✉♠❛ s♦♠❜r❛ P ′ ❞♦ ♣♦❧✐❡❞r♦ P ✳ ❊ss❛ s♦♠❜r❛ P ′ t❡♠ ❝♦♥t♦r♥♦ ❞❡ ✉♠ ♣♦❧í❣♦♥♦ ❝♦♥✈❡①♦ K′ s♦♠❜r❛ ❞❡ ✉♠ ♣♦❧✐❣♦♥❛❧ ❢❡❝❤❛❞❛ K✱ ❛ q✉❛❧ ❝❤❛♠❛r❡♠♦s ❞❡ ❝♦♥t♦r♥♦ ❛♣❛r❡♥t❡ ❞♦ ♣♦❧✐❡❞r♦ P ✱ ❢♦r♠❛❞❛ ♣♦r ❛r❡st❛s ❞❡ P ✳ ❈❛❞❛ ♣♦♥t♦ ❞♦ ♣♦❧í❣♦♥♦ K′ é ✉♠ ú♥✐❝♦ ♣♦♥t♦ ❞♦ ♣♦❧✐❡❞r♦ P ✱ ❥á ❛ s♦♠❜r❛ P ′ é s♦♠❜r❛ ❞❡ ❡①❛t❛♠❡♥t❡ ❞♦✐s ♣♦♥t♦s ❞❡ P ✳ ❯♠ ❞❡ss❡s ♣♦♥t♦s ✜❝❛ ♥❛ r❡❣✐ã♦ ✐❧✉♠✐♥❛❞❛ ❞❡ P ✱ r❡❣✐ã♦ q✉❡ ✜❝❛ ♠❛✐s ♣ró①✐♠❛ ❞❛ ❢♦♥t❡ ❧✉♠✐♥♦s❛ ❛ q✉❛❧ ❝❤❛♠❛r❡♠♦s ❞❡ P1 t♦❞♦ ♣♦♥t♦ ❞❡ss❛ r❡❣✐ã♦ é ❝❤❛♠❛❞♦ ❞❡ ♣♦♥t♦ ✐❧✉♠✐♥❛❞♦ ❡ ♦ ♦✉tr♦ ♣♦♥t♦ ✜❝❛rá ♥❛ r❡❣✐ã♦ ♠❡♥♦s ✐❧✉♠✐♥❛❞❛ ❞❡ P ✱ r❡❣✐ã♦ q✉❡ ✜❝❛ ♠❛✐s ♣ró①✐♠❛ ❞♦ ♣❧❛♥♦ H✱ ❛ q✉❛❧ ❝❤❛♠❛r❡♠♦s ❞❡ P2 t♦❞♦ ♣♦♥t♦ ❞❡ss❛ r❡❣✐ã♦ s❡rá ❝❤❛♠❛❞♦ ❞❡ ♣♦♥t♦ s♦♠❜r✐♦✳ ❉❡♣♦✐s ❞❡ss❛s ❝♦♥s✐❞❡r❛çõ❡s ✈❛♠♦s ❝❛❧❝✉❧❛r ♥♦✈❛♠❡♥t❡ ❛ s♦♠❛ ❞♦s â♥❣✉❧♦s ✐♥✲ t❡r♥♦s ❞❡ t♦❞❛s ❛s ❢❛❝❡s ❞❡ P ✳ ❉❡✈❡♠♦s ❧❡♠❜r❛r q✉❡ ❛ s♦♠❛ ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞❡ ✉♠❛ ❢❛❝❡ ❞❡ P é ✐❣✉❛❧ s♦♠❛ â♥❣✉❧♦s ✐♥t❡r♥♦s ❞❛ s♦♠❜r❛ ❞❡ss❛ ❢❛❝❡✳ ❆ss✐♠ t❡♠♦s✱ V = V0 + V1 + V2 ♦♥❞❡ V0✱ V1✱ V2 sã♦ r❡s♣❡❝t✐✈❛♠❡♥t❡ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ✈ért✐❝❡s ❞♦ ❝♦♥t♦r♥♦ ❛♣❛r❡♥t❡✱ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ✈ért✐❝❡s ❞❡ P1 ❡ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ✈ért✐❝❡s ❞❡ P2✳ P❛r❛ ❝❛❧❝✉❧❛r ❛ s♦♠❛ ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞♦ ♣♦❧✐❡❞r♦ P t❡♠♦s Si = S1 + S2✱ ♦♥❞❡ S1 é ❛ s♦♠❛ ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞❛ r❡❣✐ã♦ ✐❧✉♠✐♥❛❞❛ ❡ S2 é ❛ s♦♠❛ ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞❛ r❡❣✐ã♦ s♦♠❜r✐❛✳ ❉❛❞❛s ♣♦r ❆♥❛❧♦❣❛♠❡♥t❡ t❡♠♦s S1 = 2πV1 + π(V0 − 2). ❊♥tã♦ S2 = 2πV2 + π(V0 − 2). Si = S1 + S2 = [2πV1 + π(V0 − 2)] + [2πV2 + π(V0 − 2)] = 2πV1 + 2πV2 + π(V0 − 2) + π(V0 − 2) = 2πV1 + 2πV2 + 2π(V0 − 2) = 2π(V1 + V2 + V0 − 2). ❖r❛✱ V1 + V2 + V0 = V ✳ ❉❛í t❡♠♦s Si = 2π(V − 2). ❯s❛♥❞♦ ❛s ❡q✉❛çõ❡s ✭✸✳✶✮ ❡ ✭✸✳✷✮✿ ✶✺ ✭✸✳✷✮

(29) Si = Si 2π(A − F ) = 2π(V − 2) A−F = V −2 V − A + F = 2, ❞❡ss❛ ❢♦r♠❛ ❝♦♥❝❧✉í♠♦s ❛ ♣r♦✈❛✳ ✶✻

(30) ❈❛♣ít✉❧♦ ✹ P♦❧✐❡❞r♦s r❡❣✉❧❛r❡s ◆❡st❡ ❝❛♣ít✉❧♦ s❡rã♦ ❞❡✜♥✐❞♦s ♦s ♣♦❧✐❡❞r♦s r❡❣✉❧❛r❡s ✭♦✉ ♣♦❧✐❡❞r♦s ❞❡ P❧❛tã♦✮ ❡ ❛ ❞❡♠♦♥str❛çã♦ ❞❛ ❡①✐stê♥❝✐❛ ❞❡ s♦♠❡♥t❡ ❝✐♥❝♦ ❞❡❧❡s✳ ❆❜❛✐①♦ s❡❣✉❡♠ ❞✉❛s ❞❡✜♥✐çõ❡s ❞❡ ♣♦❧✐❡❞r♦s r❡❣✉❧❛r❡s✱ ❛ ♣r✐♠❡✐r❛ ♣❛r❛ ❚✐③③✐♦tt✐ ❬✶✾❪ ♣á❣✐♥❛ ✸✹✼ ❡ ❛ s❡❣✉♥❞❛ ❞❡ P❛❧❤❛r❡s ❬✶✹❪ ♣á❣✐♥❛ ✸✵✻✳ ❉❡✜♥✐çã♦ ✹✳✶ ➱ ♦ ♣♦❧✐❡❞r♦ ♥♦ q✉❛❧ ❛s ❢❛❝❡s sã♦ ♣♦❧í❣♦♥♦s r❡❣✉❧❛r❡s ❝♦♥❣r✉❡♥t❡s ❡ ♦s â♥❣✉❧♦s ♣♦❧✐é❞r✐❝♦s sã♦ ❝♦♥❣r✉❡♥t❡s✳ ❉❡✜♥✐çã♦ ✹✳✷ ➱ ✉♠ ♣♦❧✐❡❞r♦ ❝♦♥✈❡①♦ ❝✉❥❛s ❢❛❝❡s sã♦ ♣♦❧í❣♦♥♦s r❡❣✉❧❛r❡s✱ t♦❞♦s ✐❣✉❛✐s ❡ ♦♥❞❡ ❡♠ ❝❛❞❛ ✉♠ ❞♦s ✈ért✐❝❡s ❝♦♥❝♦rr❡ ♦ ♠❡s♠♦ ♥ú♠❡r♦ ❞❡ ❛r❡st❛s✳ ❯♠❛ ❝❛r❛❝t❡ríst✐❝❛ é ❝♦♠✉♠ ♣❛r❛ ♦s ❞♦✐s ❛✉t♦r❡s q✉❡ sã♦ ❛s ❢❛❝❡s ❢♦r♠❛❞❛s ♣♦r ♣♦❧í❣♦♥♦s r❡❣✉❧❛r❡s✱ ♣♦ré♠ ♦ ♣r✐♠❡✐r♦ ❝♦♠♣❧❡♠❡♥t❛ s✉❛ ❞❡✜♥✐çã♦ ❝✐t❛♥❞♦ â♥❣✉❧♦s ♣♦❧✐é❞r✐❝♦s✱ q✉❡ s❡❣✉♥❞♦ P❛✐✈❛ ❬✶✸❪ ♣á❣✐♥❛ ✸✽✶ sã♦ ♣♦rçõ❡s ❞♦ ❡s♣❛ç♦ ❝✉❥❛ s✉♣❡r❢í❝✐❡ é ❛ r❡✉♥✐ã♦ ❞♦s â♥❣✉❧♦s ❞❛s ❢❛❝❡s q✉❡ tê♠ ✉♠ ♠❡s♠♦ ✈ért✐❝❡✳ ❊♥q✉❛♥t♦ ♦ s❡❣✉♥❞♦ ❝✐t❛ ❛s ❛r❡st❛s q✉❡ ❝♦♥❝♦rr❡♠ ♥♦s ✈ért✐❝❡s✳ ❖s ♣♦❧✐❡❞r♦s r❡❣✉❧❛r❡s t❛♠❜é♠ sã♦ ❝♦♥❤❡❝✐❞♦s ❝♦♠♦ só❧✐❞♦s P❧❛tô♥✐❝♦s✳ ✹✳✶ ❊①✐stê♥❝✐❛ ❞❡ ❝✐♥❝♦ ♣♦❧✐❡❞r♦s r❡❣✉❧❛r❡s ❆ ❞❡♠♦♥str❛çã♦ ❛❜❛✐①♦ ❢♦✐ ❜❛s❡❛❞❛ ♥♦ ❧✐✈r♦ ❞❡ ❉❛♥t❡ ❬✻❪ ♣á❣✐♥❛ ✸✻✸✳ ❚❡♦r❡♠❛ ✹✳✶ ❊①✐t❡♠ s♦♠❡♥t❡ ❝✐♥❝♦ ♣♦❧✐❡❞r♦s r❡❣✉❧❛r❡s✳ Pr♦✈❛✿ ❱❛♠♦s ❝♦♥s✐❞❡r❛r ✉♠ ♣♦❧✐❡❞r♦ r❡❣✉❧❛r ❝♦♠ A ❛r❡st❛s✱ V ✈ért✐❝❡s ❡ F ❢❛❝❡s✱ ♦♥❞❡ n é ♦ ♥ú♠❡r♦ ❞❡ ❧❛❞♦s ❡♠ ❝❛❞❛ ❢❛❝❡ ❡ p é ♦ ♥ú♠❡r♦ ❞❡ ❛r❡st❛s q✉❡ ❝♦♥❝♦rr❡♠ ❡♠ ❝❛❞❛ ✈ért✐❝❡✳ ❯s❛♥❞♦ ❛s r❡❧❛çõ❡s ❞♦s ❡❧❡♠❡♥t♦s ❞♦s ♣♦❧✐❡❞r♦s t❡♠♦s 2A = nF = pV. ✶✼

(31) ✹✳✶✳ ❊❳■❙❚✃◆❈■❆ ❉❊ ❈■◆❈❖ P❖▲■❊❉❘❖❙ ❘❊●❯▲❆❘❊❙ ❱❛♠♦s ❡s❝r❡✈❡r A ❡ V ❡♠ ❢✉♥çã♦ ❞❡ F ✱ t❡♠♦s A = nF 2 . ❡ V = 2A. p ❈♦♠❜✐♥❛♥❞♦ ✭✹✳✶✮ ❝♦♠ ✭✹✳✷✮✱ ♦❜t❡♠♦s V = 2nF 2p = nF p . ❱❛♠♦s s✉❜st✐t✉✐r ❛s ❡q✉❛çõ❡s ✭✹✳✶✮ ❡ ✭✹✳✸✮ ♥❛ r❡❧❛çã♦ ❞❡ ❊✉❧❡r✳ ✭✹✳✶✮ ✭✹✳✷✮ ✭✹✳✸✮ 2 = V −A+F = nF p − nF 2 + F ❛ss✐♠ t❡♠♦s 2nF − pnF 2p + 2pF = 4p . 2p ❈♦❧♦❝❛♥❞♦ F ❡♠ ❡✈✐❞ê♥❝✐❛ ♥♦ ♣r✐♠❡✐r♦ ♠❡♠❜r♦ t❡♠♦s F (2n − pn + 2p) = 4p ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡ F = 4p 2n − pn + 2p . ✭✹✳✹✮ P❛r❛ q✉❡ t❡♥❤❛♠♦s ✉♠ ♥ú♠❡r♦ F ❞❡ ❢❛❝❡s ♣r❡❝✐s❛♠♦s t❡r ♦ ❞❡♥♦♠✐♥❛❞♦r 2n − pn + 2p > 0✱ ❞❛í 2n > np − 2p ❝♦❧♦❝❛♥❞♦ p ❡♠ ❡✈✐❞ê♥❝✐❛ 2n > p(n − 2) 2n ⇒ n − 2 > p. ✭✹✳✺✮ ❈♦♠♦ ✈ért✐❝❡ é ❝❛❞❛ ✉♠ ❞♦s ♣♦♥t♦s ❞❡ ✐♥t❡rs❡❝çã♦ ❞❡ ✸ ♦✉ ♠❛✐s ❛r❡st❛s✱ t❡♠♦s p ≥ 3✳ ❈♦♠♣❛r❛♥❞♦ ❝♦♠ ❛ ❡q✉❛çã♦ ✭✹✳✺✮✱ t❡♠♦s✿ ✶✽

(32) ✹✳✶✳ ❊❳■❙❚✃◆❈■❆ ❉❊ ❈■◆❈❖ P❖▲■❊❉❘❖❙ ❘❊●❯▲❆❘❊❙ 2n n−2 > p ≥ 3 ⇒ n < 6. ❈♦♠♦ ♦ ♠❡♥♦r ♥ú♠❡r♦ ❞❡ ❧❛❞♦s ❞❡ ✉♠❛ ❢❛❝❡ é três ✭❢❛❝❡ tr✐❛♥❣✉❧❛r✮✱ t❡♠♦s q✉❡ n ≥ 3✱ ♠❛s t❡♠♦s t❛♠❜é♠ q✉❡ n < 6✱ ❛ss✐♠ ♦s ✈❛❧♦r❡s ❞❡ n sã♦ ✸✱ ✹ ❡ ✺✳ ❱❛♠♦s s✉❜st✐t✉✐r ♦s ✈❛❧♦r❡s ❞❡ n ♥❛ ❡q✉❛çã♦ ✭✹✳✹✮ P❛r❛ n = 3 ♦❜t❡♠♦s F = 4p 2 · 3 − 3p + 2p = 6 4p − p . ✭✹✳✻✮ ❖❜s❡r✈❛♥❞♦ ❛ ❡q✉❛çã♦ ✭✹✳✻✮ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ p < 6 ✭♣❛r❛ s❛t✐s❢❛③❡r ❛ ❡q✉❛✲ çã♦✮✱ ❝♦♠♦ p é ♦ ♥ú♠❡r♦ ❞❡ ❛r❡st❛s q✉❡ ❝♦♥❝♦rr❡♠ ❡♠ ❝❛❞❛ ✈ért✐❝❡ ❡ p ≥ 3✱ t❡♠♦s 3 ≤ p < 6✳ P❛r❛ p = 3 s❡❣✉❡ q✉❡ ♦✉ s❡❥❛✱ F = 4·3 6−3 F = 4. ❈♦♠♦ n = 3 ✭❢❛❝❡s tr✐❛♥❣✉❧❛r❡s✮✱ F = 4 ✭q✉❛tr♦ ❢❛❝❡s ✐❣✉❛✐s✮ ❡ p = 3 ✭❡♠ ❝❛❞❛ ✈ért✐❝❡ ❝♦♥❝♦rr❡♠ três ❛r❡st❛s✮ t❡♠♦s ♦ t❡tr❛❡❞r♦✳ ❋✐❣✉r❛ ✹✳✶✿ ❚❡tr❛❡❞r♦ ✭❡sq✉❡r❞❛✮ ❡ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ♣❧❛♥✐✜❝❛❞❛ ✭❞✐r❡✐t❛✮✳ P❛r❛ p = 4 s❡❣✉❡ q✉❡ F = 4·4 6−4 ✶✾

(33) ✹✳✶✳ ❊❳■❙❚✃◆❈■❆ ❉❊ ❈■◆❈❖ P❖▲■❊❉❘❖❙ ❘❊●❯▲❆❘❊❙ t❡♠♦s F = 8. ❈♦♠♦ n = 3 ✭❢❛❝❡s tr✐❛♥❣✉❧❛r❡s✮✱ F = 8 ✭♦✐t♦ ❢❛❝❡s ✐❣✉❛✐s✮ ❡ p = 4 ✭❡♠ ❝❛❞❛ ✈ért✐❝❡ ❝♦♥❝♦rr❡♠ q✉❛tr♦ ❛r❡st❛s✮ t❡♠♦s ♦ ♦❝t❛❡❞r♦✳ ❋✐❣✉r❛ ✹✳✷✿ ❖❝t❛❡❞r♦ ✭❡sq✉❡r❞❛✮ ❡ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ♣❧❛♥✐✜❝❛❞❛ ✭❞✐r❡✐t❛✮✳ P❛r❛ p = 5 ♦❜t❡♠♦s ❞❡ss❛ ❢♦r♠❛ F = 4·5 6−5 F = 20. ❈♦♠♦ n = 3 ✭❢❛❝❡s tr✐❛♥❣✉❧❛r❡s✮✱ F = 20 ✭✈✐♥t❡ ❢❛❝❡s ✐❣✉❛✐s✮ ❡ p = 5 ✭❡♠ ❝❛❞❛ ✈ért✐❝❡ ❝♦♥❝♦rr❡♠ ❝✐♥❝♦ ❛r❡st❛s✮ t❡♠♦s ♦ ✐❝♦s❛❡❞r♦✳ ❋✐❣✉r❛ ✹✳✸✿ ■❝♦s❛❡❞r♦ ✭❡sq✉❡r❞❛✮ ❡ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ♣❧❛♥✐✜❝❛❞❛ ✭❞✐r❡✐t❛✮✳ ❆❣♦r❛ ✈❛♠♦s ✈❡r✐✜❝❛r ♦ ✈❛❧♦r ❞❡ n = 4✳ ✷✵

(34) ✹✳✶✳ ❊❳■❙❚✃◆❈■❆ ❉❊ ❈■◆❈❖ P❖▲■❊❉❘❖❙ ❘❊●❯▲❆❘❊❙ P❛r❛ n = 4 ♦❜t❡♠♦s F = 2 · 4p 4 − 4p + 2p = 8 4p − 2p . ✭✹✳✼✮ ❖❜s❡r✈❛♥❞♦ ❛ ❡q✉❛çã♦ ✭✹✳✼✮ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ p < 6 ✭♣❛r❛ s❛t✐s❢❛③❡r ❛ ❡q✉❛✲ çã♦✮✱ ❝♦♠♦ p é ♦ ♥ú♠❡r♦ ❞❡ ❛r❡st❛s q✉❡ ❝♦♥❝♦rr❡♠ ❡♠ ❝❛❞❛ ✈ért✐❝❡ ❡ p ≥ 3✱ t❡♠♦s 3 ≤ p < 6✱ ♣♦ré♠ ♦s ✈❛❧♦r❡s p = 4 ❡ p = 5 ♥ã♦ s❛t✐s❢❛③❡♠ ❛ ❡q✉❛çã♦ F = 8 4p − 2p ✱ ♣♦✐s t♦r♥❛ F ♥❡❣❛t✐✈♦✱ ❛ss✐♠ ✈❛♠♦s ✉t✐❧✐③❛r ❛♣❡♥❛s ♣❛r❛ p = 3✳ P❛r❛ p = 3 ♦❜t❡♠♦s ♦✉ s❡❥❛✱ F = 4·3 8−6 F = 6. ❈♦♠♦ n = 4 ✭❢❛❝❡s q✉❛❞r❛♥❣✉❧❛r❡s✮✱ F = 6 ✭s❡✐s ❢❛❝❡s ✐❣✉❛✐s✮ ❡ p = 3 ✭❡♠ ❝❛❞❛ ✈ért✐❝❡ ❝♦♥❝♦rr❡♠ três ❛r❡st❛s✮ t❡♠♦s ♦ ❤❡①❛❡❞r♦✳ ❋✐❣✉r❛ ✹✳✹✿ ❍❡①❛❡❞r♦ ✭❡sq✉❡r❞❛✮ ❡ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ♣❧❛♥✐✜❝❛❞❛ ✭❞✐r❡✐t❛✮✳ ❊ ♣♦r ✜♠ ✈❛♠♦s ✈❡r✐✜❝❛r ♦ ✈❛❧♦r ❞❡ n = 5✳ P❛r❛ n = 5 F = 2 · 4p 5 − 5p + 2p = 10 4p − 3p . ✷✶ ✭✹✳✽✮

(35) ✹✳✶✳ ❊❳■❙❚✃◆❈■❆ ❉❊ ❈■◆❈❖ P❖▲■❊❉❘❖❙ ❘❊●❯▲❆❘❊❙ ❆ss✐♠ ❝♦♠♦ ♣❛r❛ n = 4✱ t❡♠♦s q✉❡ ❞❛ ❡q✉❛çã♦ ✭✹✳✽✮✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ p < 6 ✭♣❛r❛ s❛t✐s❢❛③❡r ❛ ❡q✉❛çã♦✮✱ ❝♦♠♦ p é ♦ ♥ú♠❡r♦ ❞❡ ❛r❡st❛s q✉❡ ❝♦♥❝♦rr❡♠ ❡♠ ❝❛❞❛ ✈ért✐❝❡ ❡ p ≥ 3✱ t❡♠♦s 3 ≤ p < 6✱ ♣♦ré♠ ♦s ✈❛❧♦r❡s p = 4 ❡ p = 5 ♥ã♦ s❛t✐s❢❛③❡♠ ❛ ❡q✉❛çã♦ F = 10 4p − 3p ✱ ♣♦✐s t♦r♥❛ F ♥❡❣❛t✐✈♦✱ ❛ss✐♠ ✈❛♠♦s t❡st❛r ❛♣❡♥❛s ♣❛r❛ p = 3✳ P❛r❛ p = 3 s❡❣✉❡ q✉❡ t❡♠♦s F = 4 10 ·3 −9 F = 12. ❈♦♠♦ n = 5 ✭❢❛❝❡s ♣❡♥t❛❣♦♥❛✐s✮✱ ✈ért✐❝❡ ❝♦♥❝♦rr❡♠ três ❛r❡st❛s✮ t❡♠♦s F ♦ ❞=♦❞12❡❝✭❛❞❡♦❞③❡r♦❢✳❛❝❡s ✐❣✉❛✐s✮ ❡ p = 3 ✭❡♠ ❝❛❞❛ ❋✐❣✉r❛ ✹✳✺✿ ❉♦❞❡❝❛❡❞r♦ ✭❡sq✉❡r❞❛✮ ❡ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ♣❧❛♥✐✜❝❛❞❛ ✭❞✐r❡✐t❛✮✳ ❆ss✐♠ ♣r♦✈❛♠♦s ❛ ❡①✐stê♥❝✐❛ ❞❡ ❛♣❡♥❛s ❝✐♥❝♦ ♣♦❧✐❡❞r♦s r❡❣✉❧❛r❡s✳ ❆ t❛❜❡❧❛ ❛❜❛✐①♦ ♠♦str❛ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❛r❡st❛s (A)✱ ✈ért✐❝❡s (V )✱ ❢❛❝❡s (F )✱ ♦ ♥ú♠❡r♦ ❞❡ ❛r❡st❛s q✉❡ ❝♦♥❝♦rr❡ ❡♠ ❝❛❞❛ ✈ért✐❝❡ (p) ❡ ♦ ♥ú♠❡r♦ ❞❡ ❛r❡st❛s ❡♠ ❝❛❞❛ ❢❛❝❡ (n)✳ P♦❧✐❡❞r♦ ❆ ❱ ❋ ♣ ♥ ❚❡tr❛❡❞r♦ ✻ ✹ ✹ ✸ ✸ ❍❡①❛❡❞r♦ ✶✷ ✽ ✻ ✸ ✹ ❖❝t❛❡❞r♦ ✶✷ ✻ ✽ ✹ ✸ ❉♦❞❡❝❛❡❞r♦ ✸✵ ✷✵ ✶✷ ✸ ✺ ■❝♦s❛❡❞r♦ ✸✵ ✶✷ ✷✵ ✺ ✸ ❚❛❜❡❧❛ ✹✳✶✿ P♦❧✐❡❞r♦s ✷✷

(36) ✹✳✷✳ ➪❘❊❆❙ ❊ ❱❖▲❯▼❊❙ ✹✳✷ ➪r❡❛s ❡ ❱♦❧✉♠❡s ◆❡ss❛ s❡çã♦ ✈❛♠♦s ❝❛❧❝✉❧❛r ♦s ✈❛❧♦r❡s ❞❛s ár❡❛s ❞♦s ♣♦❧✐❡❞r♦s r❡❣✉❧❛r❡s ❡ ✈♦❧✉♠❡s ❞❛s r❡❣✐õ❡s ❞♦ ❡s♣❛ç♦ ❞❡❧✐♠✐t❛❞❛s ♣❡❧♦s ♣♦❧✐❡❞r♦s r❡❣✉❧❛r❡s ❡♠ ❢✉♥çã♦ ❞❡ ✉♠❛ ❛r❡st❛ α✳ ✹✳✷✳✶ ➪r❡❛s P❛r❛ ❝❛❧❝✉❧❛r ❛s ár❡❛s ✭❞❛s s✉♣❡r❢í❝✐❡s✮ ❞♦s ♣♦❧✐❡❞r♦s ❞❡ ❢❛❝❡s tr✐❛♥❣✉❧❛r❡s ❜❛st❛ ❝❛❧❝✉❧❛r ❛ ár❡❛ ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❡ ❞❡♣♦✐s ♠✉❧t✐♣❧✐❝❛r ♣❡❧♦ ♥ú♠❡r♦ ❞❡ ❢❛❝❡s✱ ❛ss✐♠ ❞❛❞♦ ✉♠ tr✐â♥❣✉❧♦ DEF ❞❡ ❧❛❞♦ α ✭❛r❡st❛ ❞♦ ♣♦❧✐❡❞r♦✮ ❡ ♣♦♥t♦ ♠é❞✐♦ M ❞♦ ❧❛❞♦ EF t❡♠♦s q✉❡ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ ♣r❡❝✐s❛♠♦s ❞❛ ❛❧t✉r❛ DM ✳ P♦❞❡♠♦s ❞❡t❡r♠✐♥á✲❧❛ ♣❡❧♦ tr✐â♥❣✉❧♦ DM F ✳ ❋✐❣✉r❛ ✹✳✻✿ ❚r✐â♥❣✉❧♦ DEF ❞❡ ❧❛❞♦ α✳ 22 2 DF ❂ MF ✰ DM ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡ DM2 ❂ DF 2 ✲ MF 2. α ❙✉❜st✐t✉✐♥❞♦ DF ♣♦r α ❡ M F ♣♦r 2 t❡♠♦s 2 DM = α2 − α2 4 ⇒ DM = 3α2 4 = α 2 √ 3. ✷✸ ✭✹✳✾✮

(37) ✹✳✷✳ ➪❘❊❆❙ ❊ ❱❖▲❯▼❊❙ P❛r❛ ❞❡t❡r♠✐♥❛r ár❡❛ ❞❛ s✉♣❡r❢í❝✐❡ S ❞♦ tr✐â♥❣✉❧♦ ❞❡ ❧❛❞♦ α✱ ❜❛st❛ ♠✉❧t✐♣❧✐❝❛r α ✭❜❛s❡✮ ♣❡❧❛ ❛❧t✉r❛ ✭✹✳✾✮ ❡ ❞✐✈✐❞✐r ♦ ♣r♦❞✉t♦ ♣♦r ✷ ❡ ♣♦rt❛♥t♦✱ S = α. α 2 √ 3 2 √ S = α2 4 3. ✭✹✳✶✵✮ ❊♥tã♦ ♣❛r❛ ❝❛❧❝✉❧❛r ❛ ár❡❛ ❞❛ s✉♣❡r❢í❝✐❡ ❞❡ ✉♠ t❡tr❛❡❞r♦✱ ♦❝t❛❡❞r♦ ❡ ✐❝♦s❛❡❞r♦ ❞❡ ❛r❡st❛ α ❜❛st❛ ♠✉❧t✐♣❧✐❝❛r ♣♦r ✹✱ ✽ ❡ ✷✵ r❡s♣❡❝t✐✈❛♠❡♥t❡ ✭♥ú♠❡r♦ ❞❡ ❢❛❝❡s✮ ♣❡❧❛ ❡q✉❛çã♦ ✭✹✳✶✵✮ ❛ss✐♠ ♣❛r❛ ❛ ár❡❛ ❞♦ t❡tr❛❡❞r♦ t❡♠♦s q✉❡ √ S = 4· α2 4 3 ❡ ♣♦rt❛♥t♦ √ S = α2 3✳ ❆ ár❡❛ ❞♦ ♦❝t❛❡❞r♦ é ❞❛❞❛ ♣♦r √ S = 8· α2 4 3 ❡ ♣♦rt❛♥t♦ √ ❙ ❂ ✷α2 3. ❊ ♣♦r ✜♠ ❛ ár❡❛ ❞♦ ✐❝♦s❛❡❞r♦ é ❞❛❞❛ ♣♦r √ S = 20 · α2 4 3 ❡ ❛ss✐♠ t❡♠♦s √ ❙ ❂ ✺α2 3. P❛r❛ ❝❛❧❝✉❧❛r ❛ ár❡❛ ❞❛ s✉♣❡r❢í❝✐❡ S ❞❡ ✉♠ ❤❡①❛❡❞r♦ ❞❡ ❧❛❞♦ α ✭❛r❡st❛ ❞♦ ♣♦❧✐❡❞r♦✮ ❜❛st❛ ❞❡t❡r♠✐♥❛r ❛ ár❡❛ ❞❡ ✉♠❛ ❢❛❝❡ ❡ ♠✉❧t✐♣❧✐❝❛r ♣♦r ✻✳ ❈♦♠♦ ❛s ❢❛❝❡s sã♦ q✉❛❞r❛❞♦s t❡♠♦s q✉❡ ❛ ár❡❛ ❞♦ q✉❛❞r❛❞♦ é ❞❛❞❛ ♣♦r S = JG · GH. ✷✹

(38) ✹✳✷✳ ➪❘❊❆❙ ❊ ❱❖▲❯▼❊❙ ❋✐❣✉r❛ ✹✳✼✿ ◗✉❛❞r❛❞♦ ❞❡ ❧❛❞♦ α✳ ❙✉❜st✐t✉✐♥❞♦ JG ❡ GH ♣♦r α t❡♠♦s S = α2. ✭✹✳✶✶✮ ❊♥tã♦ ♣❛r❛ ❝❛❧❝✉❧❛r ❛ ár❡❛ ❞❛ s✉♣❡r❢í❝✐❡ ❞❡ ✉♠ ❤❡①❛❡❞r♦ ❜❛st❛ ♠✉❧t✐♣❧✐❝❛r ♣♦r ✻ ✭♥ú♠❡r♦ ❞❡ ❢❛❝❡s✮ ❛ ❡q✉❛çã♦ ✭✹✳✶✶✮✱ ❛ss✐♠ ♣❛r❛ ❛ ár❡❛ ❞♦ ❤❡①❛❡❞r♦ t❡♠♦s ❙ ❂ ✻α2. ❊ ♣♦r ✜♠✱ ❝❛❧❝✉❧❛♥❞♦ ❛ ár❡❛ ❞❛ s✉♣❡r❢í❝✐❡ ❞❡ ✉♠ ❞♦❞❡❝❛❡❞r♦ ❞❡ ❛r❡st❛ α✳ ❈♦♠♦ s✉❛s ❢❛❝❡s sã♦ ♣❡♥tá❣♦♥♦s ✈❛♠♦s ❞✐✈✐❞í✲❧♦s ❡♠ tr✐â♥❣✉❧♦s ✐sós❝❡❧❡s ❧✐❣❛♥❞♦ ♦s ✈ért✐❝❡s ❛♦ ❝❡♥tr♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❝✐r❝✉♥s❝r✐t❛ ❞❡ r❛✐♦ ON = R ❡ ♦ r❛✐♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ✐♥s❝r✐t❛ ❞❡ r❛✐♦ OP = r✱ ♦♥❞❡ P é ♦ ♣♦♥t♦ ♠é❞✐♦ ❞❡ ✉♠ ❧❛❞♦ ❞♦ ♣❡♥tá❣♦♥♦✳ ❋✐❣✉r❛ ✹✳✽✿ P❡♥tá❣♦♥♦ JKLMN ❞❡ ❧❛❞♦ α✳ ❖❜s❡r✈❛✲s❡ q✉❡ ♦ tr✐â♥❣✉❧♦ OP N t❡♠ ❝♦♠♦ ❧❛❞♦s ♦s s❡❣♠❡♥t♦s ON ✭r❛✐♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❝✐r❝✉♥s❝r✐t❛ ❛♦ ♣❡♥tá❣♦♥♦✮✱ NP ✭♠❡t❛❞❡ ❞♦ ❧❛❞♦ ❞♦ ♣❡♥tá❣♦♥♦✮ ❡ OP ✭r❛✐♦ ❞❛ ❝✐r❝✉♥❢ê♥❝✐❛ ✐♥s❝r✐t❛ ❛♦ ♣❡♥tá❣♦♥♦✮✳ ❖ r❛✐♦ r t❛♠❜é♠ é ❝❤❛♠❛❞♦ ❞❡ ❛♣ót❡♠❛ ❞♦ ♣❡♥tá❣♦♥♦✳ ❆ ❞❡✜♥✐çã♦ ❞❡ á♣♦t❡♠❛ ❛❜❛✐①♦ é ❞❛❞❛ ♣♦r ■❡③③✐ ❬✾❪ ♣á❣✐♥❛ ✹✼✶✳ ❉❡✜♥✐çã♦ ✹✳✸ ❆♣ót❡♠❛ ❞❡ ✉♠ ♣♦❧í❣♦♥♦ r❡❣✉❧❛r é ♦ s❡❣♠❡♥t♦ q✉❡ ✉♥❡ ♦ ❝❡♥tr♦ ❞♦ ♣♦❧í❣♦♥♦ ❛♦ ♣♦♥t♦ ♠é❞✐♦ ❞❡ ✉♠ ❧❛❞♦✳ ✷✺

(39) ✹✳✷✳ ➪❘❊❆❙ ❊ ❱❖▲❯▼❊❙ ❖❜s❡r✈❛♥❞♦ ♦ tr✐â♥❣✉❧♦ OJN t❡♠♦s✿ JN é ❛ ❜❛s❡ ❞♦ tr✐â♥❣✉❧♦ ❡ OP ❛ ❛❧t✉r❛ ❞♦ tr✐â♥❣✉❧♦✱ ❛ss✐♠ ♣❛r❛ ❝❛❧❝✉❧❛r♠♦s ❛ ár❡❛ ❞❛ s✉♣❡r❢í❝✐❡ S ❞♦ tr✐â♥❣✉❧♦ OJN t❡♠♦s S = JN · OP 2 = α· 2 r . ✭✹✳✶✷✮ ❈♦♠♦ q✉❡r❡♠♦s ❝❛❧❝✉❧❛r ❛ ár❡❛ ❞❛ s✉♣❡r❢í❝✐❡ S ❞♦ ♣❡♥tá❣♦♥♦ ❞❡ ❛r❡st❛ A ♣r❡❝✐✲ s❛♠♦s ♠✉❧t✐♣❧✐❝❛r ❛ ❡q✉❛çã♦ ✭✹✳✶✷✮ ♣♦r ✺✳ S = 5 · α· 2 r . ❈♦♠♦ t❡♠♦s ❞♦③❡ ♣❡♥tá❣♦♥♦s ♥♦ ❞♦❝❡❝❛❡❞r♦ s✉❛ ár❡❛ é ❞❛❞❛ ♣♦r✿ ❙ ❂ ✸✵α · r. ❈♦♠♦ ♣♦❞❡♠♦s ♦❜s❡r✈❛r ❛ ár❡❛ ❞❛ s✉♣❡r❢í❝✐❡ ❞♦ ♣❡♥tá❣♦♥♦ ❡stá ❞❡t❡r♠✐♥❛❞❛ ♣❡❧❛ ❛r❡st❛ α ❡ ♣❡❧♦ á♣♦t❡♠❛ r ❡ss❡ ❢❛t♦ ❞❡✈❡✲s❡ ♣♦r ✜❝❛r ♠❛✐s ❝♦♠♣r❡❡♥sí✈❡❧ ❡ ❢á❝✐❧ ❞❡ ❞❡t❡r♠✐♥á✲❧❛✱ ♣♦ré♠ ❡❧❛ ♣♦❞❡r✐❛ s❡r ❡①♣r❡ss❛ ❛♣❡♥❛s ❝♦♠ ❛ ❛r❡st❛ ✳ √ ❙ ❂ ✸α2 25 + 10 5. ✹✳✷✳✷ ❱♦❧✉♠❡s ❖s ❝á❧❝✉❧♦s ❞♦s ✈♦❧✉♠❡s✱ ❛ss✐♠ ❝♦♠♦ ♦s ❞❛s ár❡❛s✱ t❛♠❜é♠ s❡rã♦ ❡♠ ❢✉♥çã♦ ❞❛ ❛r❡st❛ α✳ ❖ ✈♦❧✉♠❡ é ❛ r❡❣✐ã♦ ❧✐♠✐t❛❞❛ ♣❡❧♦ ♣♦❧✐❡❞r♦✳ ❋✐❣✉r❛ ✹✳✾✿ ❚❡tr❛❡❞r♦ ❞❡ ❛r❡st❛ α✳ ❈♦♠ ♦ t❡tr❛❡❞r♦ ❜❛st❛ ♠✉❧t✐♣❧✐❝❛r ❛ ár❡❛ ❞❛ ❜❛s❡ ♣❡❧❛ ❛❧t✉r❛ ❡ ❞✐✈✐❞✐r ♣♦r ✸✱ ♣♦✐s tr❛t❛✲s❡ ❞❡ ✉♠❛ ♣✐râ♠✐❞❡ ✭✈❡r s❡çã♦ ✹✳✸✮✳ ❏á t❡♠♦s ❛ ár❡❛ ❞❛ ❜❛s❡ ❞❛ ❡q✉❛çã♦ ✭✹✳✶✵✮✳ ✷✻

(40) ✹✳✷✳ ➪❘❊❆❙ ❊ ❱❖▲❯▼❊❙ ❆ ❛❧t✉r❛ ❞♦ ♣♦❧✐❡❞r♦ ♣♦❞❡ s❡r ❝❛❧❝✉❧❛❞❛ ✉t✐❧✐③❛♥❞♦ ♦ tr✐â♥❣✉❧♦ DGF r❡tâ♥❣✉❧♦ ❡♠ G✱ ♦♥❞❡ DG s❡❣♠❡♥t♦ q✉❡ ❝♦rr❡s♣♦♥❞❡ ❛ ❛❧t✉r❛ ❞♦ ♣♦❧✐❡❞r♦✱ DF ❂ α ❡ GF ❂ 2 3 α✱ ♣♦✐s é ♦ ❜❛r✐❝❡♥tr♦ ❞❛ ❜❛s❡✳ P❛r❛ ♣r♦✈❛r GF ❂ 2 3 α ❜❛st❛ tr❛❜❛❧❤❛r ❝♦♠ ♦ ♣♦❧í❣♦♥♦ q✉❡ ❢♦r♠❛ ❛ ❜❛s❡✳ ❈♦♠♦ ♦ tr✐â♥❣✉❧♦ é ❡q✉✐❧át❡r♦✱ ✐♥❝❡♥tr♦ ✭❡♥❝♦♥tr♦ ❞❛s ❜✐ss❡tr✐③❡s✮✱ ❜❛r✐❝❡♥tr♦ ✭❡♥❝♦♥tr♦ ❞❛s ♠❡❞✐❛♥❛s✮ ❡ ♦rt♦❝❡♥tr♦ ✭❡♥❝♦♥tr♦ ❞❛s ❛❧t✉r❛s✮ t♦❞♦s s❡ ❡♥❝♦♥tr❛♠ ♥✉♠ ú♥✐❝♦ ♣♦♥t♦✱ ♥♦ ❡①❡♠♣❧♦ ♦ ♣♦♥t♦ G✳ ❋✐❣✉r❛ ✹✳✶✵✿ ❚r✐â♥❣✉❧♦ ❞❛ ❜❛s❡ ❞♦ t❡tr❛❡❞r♦ ❞❡ ❛r❡st❛ α✳ ❉❛í t❡♠♦s q✉❡ ♥♦ tr✐â♥❣✉❧♦ ❢♦r♠❛❞♦ ♣❡❧♦s ♣♦♥t♦s F ✱ G ❡ O q✉❡ é ♣♦♥t♦ ♠é❞✐♦ ❞♦ ❧❛❞♦ F H✳ ❖ â♥❣✉❧♦ GF O ♠❡❞❡ 30 ◦ ✱ ♣♦✐s t❡♠♦s ✉♠❛ ❜✐ss❡tr✐③ ❝♦♠♦ r❡t❛ s✉♣♦rt❡ ❞♦ s❡❣♠❡♥t♦ F G✳ ❯t✐❧✐③❛♥❞♦✲s❡ ❞❛ r❡❧❛çã♦ ❝♦ss❡♥♦ α cos 30 ◦ = 2 FG √ ⇒ FG = α 3 3. 2 ❊q✉✐✈❛❧❡♥t❡ ❛ 3 ❞❛ ❛❧t✉r❛ ❞❡ ✉♠❛ ❢❛❝❡✳ ❆❣♦r❛ ✉s❛♥❞♦ ♦ tr✐â♥❣✉❧♦ DGF ♣♦❞❡♠♦s ❞❡t❡r♠✐♥❛r ❛ ❛❧t✉r❛ ❞♦ ♣♦❧✐❡❞r♦✱ ♦♥❞❡ DF 2 = DG2 + F G2 2 ⇒ DG = 22 DF − F G . √ ❖♥❞❡ s✉❜st✐t✉✐♥❞♦ ♦s ✈❛❧♦r❡s ❞❡ DF = α ❡ FG = α3 3 ❝♦♥❝❧✉í♠♦s q✉❡ √ DG = α 3 6. ✷✼

(41) ✹✳✷✳ ➪❘❊❆❙ ❊ ❱❖▲❯▼❊❙ ❆ss✐♠ ♦ ✈♦❧✉♠❡ ❞♦ t❡tr❛❡❞r♦ DEF H é √√ α2 3 α 6 V= 4 ·3 3 ♦✉ s❡❥❛✱ √ V = α2 12 . P❛r❛ ♦ ❤❡①❛❡❞r♦ ❜❛st❛ ♠✉❧t✐♣❧✐❝❛r♠♦s ❛ ár❡❛ ❞❛ ❜❛s❡ q✉❡ ❥á t❡♠♦s ❞❛ ❡q✉❛çã♦ ✭✹✳✶✶✮ ♣❡❧❛ ❛❧t✉r❛✱ q✉❡ é ♦ ♣ró♣r✐♦ α✳ ❆ss✐♠ ♦ ✈♦❧✉♠❡ ♣❛r❛ ✉♠ ❤❡①❛❡❞r♦ ❞❡ ❛r❡st❛ α t❡♠♦s V = α2 · α ♦✉ s❡❥❛✱ V = α3. P❛r❛ ♦ ♦❝t❛❡❞r♦ ♣♦❞❡♠♦s ♦❜s❡r✈❛r q✉❡ ❡❧❡ é ❢♦r♠❛❞♦ ♣♦r ❞✉❛s ♣✐râ♠✐❞❡s ❞❡ ❜❛s❡ q✉❛❞r❛❞❛✱ ❛ss✐♠ ♣r❡❝✐s❛♠♦s ❝❛❧❝✉❧❛r ♦ ✈♦❧✉♠❡ ❞❡ ❛♣❡♥❛s ✉♠❛ ❡ ❧♦❣♦ ❡♠ s❡❣✉✐❞❛ ♠✉❧t✐♣❧✐❝❛r♠♦s ♣♦r ✷✳ ❈♦♠♦ ❛ ❜❛s❡ é q✉❛❞r❛❞❛ t❡♠♦s s✉❛ ár❡❛ ❞❛ ❡q✉❛çã♦ ✭✹✳✶✶✮✱ ♣r❡❝✐s❛♠♦s ❛♣❡♥❛s ❞❛ ❛❧t✉r❛ ❞❛ ♣✐râ♠✐❞❡✱ ❥á s❛❜❡♠♦s ❛ ❛❧t✉r❛ ❞❡ ✉♠❛ ❢❛❝❡ tr✐❛♥❣✉❧❛r r❡❣✉❧❛r ❞❡ ❛r❡st❛ α ♣❡❧❛ ❡q✉❛ç ❛♦ ✭✹✳✾✮ ❡ ♦ ❛♣ót❡♠❛ ❞❛ ❜❛s❡ GM s❡rá ❛ ♠❡t❛❞❡ ❞❛ ❛r❡st❛ α✳ ❋✐❣✉r❛ ✹✳✶✶✿ ❖❝t❛❡❞r♦ ❞❡ ❛r❡st❛ α✳ ✷✽

(42) ✹✳✷✳ ➪❘❊❆❙ ❊ ❱❖▲❯▼❊❙ P❛r❛ ❝❛❧❝✉❧❛r DGM ♦♥❞❡ DM ❢♦r♠❛ ❛=❛❧αt2✉√r❛3 ❞❡ ❡ ✉♠❛ GM ❞❛s ♣✐râ♠✐❞❡s ♣♦❞❡♠♦s tr❛❜❛❧❤❛r = α 2 ✱ ❛ss✐♠ ♣♦❞❡♠♦s ❞❡t❡r♠✐♥❛r ❝♦♠ ♦ tr✐â♥❣✉❧♦ DG ❞❛ s❡❣✉✐♥t❡ 2 DG = 2 DM − 2 GM ⇒ 2 DG = α23 4 − α2 4 ❧♦❣♦✱ √ DG = α 2 2. ❆ss✐♠ ♦ ✈♦❧✉♠❡ ❞❡ ✉♠❛ ♣✐râ♠✐❞❡ é √ Vpir = α2 α ·2 3 2 ❞❛í ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ ♣❛r❛ ♦ ✈♦❧✉♠❡ ❞♦ ♦❝t❛❡❞r♦ t❡♠♦s √ V = 2· α3 6 2 ♦✉ s❡❥❛✱ √ V = α3 3 2. ❆❣♦r❛ ✈❛♠♦s ❝❛❧❝✉❧❛r ♦ ✈♦❧✉♠❡ ❞❡ ✉♠ ❞♦❞❡❝❛❡❞r♦ ❞❡ ❛r❡st❛ α✱ ❛❧❣✉♠❛s ✐♥❢♦r✲ ♠❛çõ❡s ❢♦r❛♠ ✉t✐❧✐③❛❞❛s ♣♦r ❙ér❣✐♦✱ ✭✈❡❥❛ ❬✶✻❪✮✳ Pr✐♠❡✐r❛♠❡♥t❡ ✈❛♠♦s ❞✐✈✐❞✐r ♦ ❞♦❞❡❝❛❡❞r♦ ❡♠ ♦✉tr♦s ♣♦❧✐❡❞r♦s ❝♦♥❢♦r♠❡ ✐❧✉str❛ ❛ ✜❣✉r❛ ❋✐❣✉r❛ ✹✳✶✷✿ ❉♦❞❡❝❛❡❞r♦ ❞❡❝♦♠♣♦st♦ ❡♠ ♦✉tr♦s ♣♦❧í❣♦♥♦s✳ ❖ ❞♦❞❡❝❛❡❞r♦ ✜❝❛ ❞✐✈✐❞✐❞♦ ❡♠ ✉♠ ❝✉❜♦ ❡ s❡✐s ♦✉tr♦s ♣♦❧✐❡❞r♦s ❝♦♥❣r✉❡♥t❡s✱ ♣r✐♠❡✐r❛♠❡♥t❡ ✈❛♠♦s ❞❡t❡r♠✐♥❛r ❛ ❛r❡st❛ ❞♦ ❝✉❜♦✳ ❖❜s❡r✈❛♥❞♦ ❛ ❞✐✈✐sã♦ ❞♦ ❞♦❞❡✲ ❝❛❡❞r♦ ♣♦❞❡♠♦s ♣❡r❝❡❜❡r q✉❡ ♦ ♣❡♥tá❣♦♥♦ é ❞✐✈✐❞✐❞♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✳ ❆ r❡t❛ s s❡❝❝✐♦♥❛ ♦ ♣❡♥tá❣♦♥♦ ♥♦ s❡❣♠❡♥t♦ KM ♥♦ q✉❛❧ t♦r♥❛✲s❡ ❛r❡st❛ ❞♦ ❝✉❜♦✳ P❛r❛ ❞❡t❡r✲ ♠✐♥❛r KM tr❛ç❛♠♦s ✉♠❛ ♣❡r♣❡♥❞✐❝✉❧❛r t ♣❡❧♦ ♣♦♥t♦ L ♦♥❞❡ t❡r❡♠♦s ✉♠ tr✐â♥❣✉❧♦ ✐sós❝❡❧❡s KLM✳ ❈♦♠♦ ♦ â♥❣✉❧♦ K t❡♠ ♠❡❞✐❞❛ ✐❣✉❛❧ ❛ 108 ◦ ✭â♥❣✉❧♦s ❞♦ ♣❡♥tá❣♦♥♦✮ ✷✾

(43) ✹✳✷✳ ➪❘❊❆❙ ❊ ❱❖▲❯▼❊❙ ❋✐❣✉r❛ ✹✳✶✸✿ ❘❡t❛ s s❡❝❝✐♦♥❛♥❞♦ ♦ ♣❡♥tá❣♦♥♦ ❏❑▲▼◆✳ ❡ s❛❜❡♥❞♦ q✉❡ ♦ tr✐â♥❣✉❧♦ KLM é ✐sós❝❡❧❡s ❝♦♥❝❧✉í♠♦s q✉❡ ♦s ♦✉tr♦s ❞♦✐s â♥❣✉❧♦s sã♦ ❝♦♥❣r✉❡♥t❡s✱ ❝❛❞❛ ✉♠ ♠❡❞✐♥❞♦ 36 ◦ ✳ ❆❣♦r❛ ♦❜s❡r✈❛♥❞♦ ♦ tr✐â♥❣✉❧♦ P KL ♦♥❞❡ ❡♠ P t❡♠♦s ✉♠ â♥❣✉❧♦ r❡t♦✱ ♣♦❞❡♠♦s ❛♣❧✐❝❛r ❛ r❡❧❛çã♦ ❞♦ ❝♦ss❡♥♦ ❞♦ â♥❣✉❧♦ ❞❡ 36 ◦ ♣❛r❛ ❞❡t❡r♠✐♥❛r ♦ ✈❛❧♦r ❞❡ KP q✉❡ é ♠❡t❛❞❡ ❞❡ KM ❛ ❛r❡st❛ ❞♦ ❝✉❜♦✳ cos 36 ◦ = KP KL √ ⇒ 1 + 4 5 = KP α , ❧♦❣♦ √ KP = 1 + 4 5 · α. KM ❈♦♠♦ KP ❂ 2 s❡❣✉❡ q✉❡ √ KM = 1 + 2 5 · α. ✭✹✳✶✸✮ P❛r❛ ❢❛❝✐❧✐t❛r ❛ ♠❛♥✐♣✉❧❛çã♦ ❞♦s ❝á❧❝✉❧♦s q✉❡ s❡❣✉❡♠✳ ❱❛♠♦s ❢❛③❡r ❛ s❡❣✉✐♥t❡ s✉❜st✐t✉✐çã♦ √ 1+ 2 5 =β ✭✹✳✶✹✮ s❡❣✉❡ q✉❡ KM = β · α. ✭✹✳✶✺✮ ✸✵

(44) ✹✳✷✳ ➪❘❊❆❙ ❊ ❱❖▲❯▼❊❙ ❆❣♦r❛ ♣r❡❝✐s❛♠♦s ❞❡t❡r♠✐♥❛r ♦ ✈♦❧✉♠❡ ❞♦s ♦✉tr♦s ♣♦❧✐❡❞r♦s✱ ♣❛r❛ ♠❡❧❤♦r❛r ❛ ❝♦♠♣r❡❡♥sã♦ ✈❛♠♦s ❞❡t❡r♠✐♥❛r ❛♣❡♥❛s ♦ ✈♦❧✉♠❡ ❞❡ ✉♠ ❞♦s ♣♦❧✐❡❞r♦s✱ ♦ r❡s✉❧t❛❞♦ ♣♦❞❡♠♦s ♠✉❧t✐♣❧✐❝❛r ♣♦r ✻✳ ❱❛♠♦s ❞✐✈✐❞í✲❧♦s ❡♠ ♦✉tr♦s ♣♦❧✐❡❞r♦s✳ ❯♠ ♣r✐s♠❛ r❡t♦ ❡ ❛s ❞✉❛s ♣❛rt❡s ❧❛t❡r❛✐s q✉❡ ❥✉♥t❛s ❢♦r♠❛♠ ✉♠❛ ♣✐râ♠✐❞❡✳ ❆❣♦r❛ ♣r❡❝✐s❛♠♦s ❡s❝r❡✈❡r ♦ ✈♦❧✉♠❡ ❞❡ss❡s ♣♦❧í❣♦♥♦s ❡♠ ❢✉♥çã♦ ❞❛ ❛r❡st❛ ❞♦ ❞♦❞❡❝❛❡❞r♦ α✳ ❱❛♠♦s ❞❡t❡r♠✐♥❛r JT ✭♦✉ NU✮ ❛❧t✉r❛ ❞❛ ♣✐râ♠✐❞❡ ❞❡ ❜❛s❡ IKMO✳ ❋✐❣✉r❛ ✹✳✶✹✿ Pr✐s♠❛ r❡t♦ ♥♦ ♠❡✐♦ ❡ ❛ ❥✉♥çã♦ ❞♦s ♣♦❧✐❡❞r♦s ❞❛s ❧❛t❡r❛✐s ❢♦r♠❛♠ ✉♠❛ ♣✐râ♠✐❞❡✳ ❆♥t❡s ✈❛♠♦s ❡s❝r❡✈❡r ♦s s❡❣♠❡♥t♦s ❡♥✈♦❧✈✐❞♦s ♣♦r ❧❡tr❛s✱ ❛♣❡♥❛s ♣❛r❛ ❢❛❝✐❧✐t❛r ♦s ❝á❧❝✉❧♦s✱ s❡❣✉❡ q✉❡ KM ❂ c✱ KR ❂ SM ❂ b✱ RS ❂ JN ❂ α✱ JT ❂ NU ❂ h✱ JR ❂ JQ ❂ NS ❂ P N ❂ d✳ ❉❛í t❡♠♦s b = c − 2 α . ✭✹✳✶✻✮ α2 = b2 + d2. ✭✹✳✶✼✮ ❉❛ ❡q✉❛çã♦ ✭✹✳✶✼✮ ✐s♦❧❛♥❞♦ d2 ❡ s✉❜st✐t✉✐♥❞♦ b ♣♦r ✭✹✳✶✻✮ ✜❝❛♠♦s ❝♦♠ d2 = α2 − b2 = α2 − c−α 2 . 2 ❆❣♦r❛ ✈❛♠♦s ✉t✐❧✐③❛r ❛ ❡q✉❛çã♦ d2 = c 2 2 + h2. ■s♦❧❛♥❞♦ h2 ❡ s✉❜st✐t✉✐♥❞♦ ♦ ✈❛❧♦r ❞❛ ❡q✉❛çã♦ ✭✹✳✶✽✮ ✭✹✳✶✾✮ h2 = α2 − c−α 2 2− c2 2 ❞❡s❡♥✈♦❧✈❡♥❞♦ ❛s ♣♦tê♥❝✐❛s ♥♦ s❡❣✉♥❞♦ ♠❡♠❜r♦ ❡ ❛❣r✉♣❛♥❞♦ ♦s t❡r♠♦s s❡♠❡✲ ❧❤❛♥t❡s ✜❝❛♠♦s ❝♦♠ h2 = 3 · α2 −2 · c2 4 + 2 · α · c, ✸✶

(45) ✹✳✷✳ ➪❘❊❆❙ ❊ ❱❖▲❯▼❊❙ ♠❛s c = KM ❡ ♣❡❧❛ ❡q✉❛çã♦ ✭✹✳✶✺✮ t❡♠♦s q✉❡ h2 = 3 · α2 − 2 · (β · α)2 4 + 2 · α · β · α ⇒ h2 = α2 4 · [3 − 2 · β(β − 1)] ♦♥❞❡ ♣♦❞❡♠♦s s✉❜st✐t✉✐r β ✈❡r✐✜❝❛♥❞♦ ❛ ❡q✉❛çã♦ ✭✹✳✶✹✮✳ ❉❡♣♦✐s ❛❣r✉♣❛♥❞♦ ♦s t❡r♠♦s s❡♠❡❧❤❛♥t❡s ❡ r❡s♦❧✈❡♥❞♦ ❛s ♦♣❡r❛çõ❡s ❝❤❡❣❛♠♦s ❛ h = α 2. ✭✹✳✷✵✮ ❈♦♠♦ t❡♠♦s ♦ ✈❛❧♦r ❞❛ ❛❧t✉r❛ h ♣♦❞❡♠♦s ❝❛❧❝✉❧❛r ♦s ✈❛❧♦r❡s ❞♦ ✈♦❧✉♠❡ ❞♦ ❝✉❜♦ V1✱ ❞♦s s❡✐s ♣r✐s♠❛s V2 ❡ ❞❛s s❡✐s ♣✐râ♠✐❞❡s V3 ❡♠ ❢✉♥çã♦ ❞❛ ❛r❡st❛ α q✉❡ ❢♦r♠❛♠ ♦ ❞♦❞❡❝❛❡❞r♦✳ ❆ss✐♠ ♦ ✈♦❧✉♠❡ ❞♦ ❞♦❞❡❝❛❡❞r♦ V s❡rá ✐❣✉❛❧ ❛ V = V1 + 6 · V2 + 6 · V1. ✭✹✳✷✶✮ ❊s❝r❡✈❡♥❞♦ ♦s ✈♦❧✉♠❡s ❞♦s ♦✉tr♦s ♣♦❧✐❡❞r♦s ❡♠ ❢✉♥çã♦ ❞♦ ✈❛❧♦r ❞❡ α ❡ β t❡♠♦s q✉❡ V1 = (α · β)3 ✭✹✳✷✷✮ V2 = α3 · 4 β ✭✹✳✷✸✮ V1 = α3 · β(β 6 − 1) ✭✹✳✷✹✮ ❙✉❜st✐t✉✐♥❞♦ ♥❛ ❡q✉❛çã♦ ✭✹✳✷✶✮ ♦s ✈❛❧♦r❡s ❞❡ V1✱ V2 ❡ V3 ❞❛s ❡q✉❛çõ❡s ✭✹✳✷✷✮✱ ✭✹✳✷✸✮ ❡ ✭✹✳✷✹✮✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ✜❝❛r❡♠♦s ❝♦♠ V = (α · β)3 + 6 · α3 · 4 β + 6 · α3 · β(β 6 − 1) ❛ss✐♠ t❡♠♦s q✉❡ ⇒ V = α3 · β3 + 3 · α3 2 · β + α3 · β(β − 1) V = α3 · β β2 + β + 1 2 . P❛r❛ ✜♥❛❧✐③❛r ❜❛st❛ s✉❜st✐t✉ír♠♦s ♦ ✈❛❧♦r ❞❡ β ❞❛ ❡q✉❛çã♦ ✭✹✳✶✹✮ √ √2 √ V = α3 · 1+ 5 2 · 1+ 2 5 + 1 + 2 5 + 1 2 . r❡s♦❧✈❡♥❞♦ ❛s ♦♣❡r❛çõ❡s ❡ ❛❣r✉♣❛♥❞♦ ♦s t❡r♠♦s s❡♠❡❧❤❛♥t❡s t❡♠♦s q✉❡ ♦ ✈♦❧✉♠❡ ❞❡ ✉♠ ❞♦❞❡❝❛❡❞r♦ ❡♠ ❢✉♥çã♦ ❞❛ ❛r❡st❛ α✳ ✸✷

(46) ✹✳✷✳ ➪❘❊❆❙ ❊ ❱❖▲❯▼❊❙ √ V = α3 15 + 7 15 4 . ❊ ♣♦r ✜♠ ✈❛♠♦s ❞❡t❡r♠✐♥❛r ♦ ✈♦❧✉♠❡ ❞❡ ✉♠ ✐❝♦s❛❡❞r♦ ❞❡ ❛r❡st❛ α✱ ❛❧❣✉♠❛s ♣❛r✲ t❡s ❞❛ ❞❡♠♦♥str❛çã♦ ❢♦r❛♠ ❜❛s❡❛❞❛s ❡♠ ●r❛♥❥❛ ❡ ❈♦st❛ ✭✈❡❥❛ ❬✽❪✮✳ P❛r❛ t❛❧ ✈❛♠♦s ❞✐✈✐❞✐r ♦ ✐❝♦s❛❡❞r♦ ❡♠ ✷✵ ♣✐râ♠✐❞❡s ❞❡ ❜❛s❡s ❢♦r♠❛❞❛s ♣❡❧❛s ❢❛❝❡s ❞♦ ✐❝♦s❛❡❞r♦ ✭tr✐✲ â♥❣✉❧♦s ❡q✉✐❧át❡r♦s✮ ❡ ❛r❡st❛s ❧❛t❡r❛✐s q✉❡ t♦❝❛♠ ♥♦ ♣♦♥t♦ C ♥♦ ❝❡♥tr♦ ❞♦ ✐❝♦s❛❡❞r♦✱ ❡ss❡ ♣♦♥t♦ ❡stá ♥❛ ♠❛✐♦r ❞✐❛❣♦♥❛❧ ❞♦ ✐❝♦s❛❡❞r♦ ❡ ❞✐✈✐❞❡ ❡ss❛ ❞✐❛❣♦♥❛❧ ❛♦ ♠❡✐♦✳ ❋✐❣✉r❛ ✹✳✶✺✿ ❉✐✈✐❞✐♥❞♦ ♦ ✐❝♦s❛❡❞r♦ ❡♠ ✷✵ ♣✐râ♠✐❞❡s✳ P❛r❛ ✐ss♦ ✈❛♠♦s ♦❜s❡r✈❛r ❛ ♣♦❧✐❣♦♥❛❧ KP QRMS ❢♦r♠❛❞❛ ♣❡❧❛s ❛❧t✉r❛s h ❞❛s ❢❛❝❡s ❡ ❛r❡st❛s α ❡ ♦ ♣❡♥tá❣♦♥♦ r❡❣✉❧❛r JKLMN✳ ❏á t❡♠♦s ♦ ✈❛❧♦r ❞♦ s❡❣♠❡♥t♦ KM ❞❛ ❡q✉❛çã♦ ✭✹✳✶✸✮✳ ❋✐❣✉r❛ ✹✳✶✻✿ P♦❧✐❣♦♥❛❧ KP QRMS ❡ ♣❡♥tá❣♦♥♦ r❡❣✉❧❛r JKLMN✳ ❆ss✐♠ ♣❡❧❛ ♣♦❧✐❣♦♥❛❧ KP QRMS ♣♦❞❡♠♦s ❝❛❧❝✉❧❛r ♦ ✈❛❧♦r ❞❛ ❞✐❛❣♦♥❛❧ d ❞♦ ✐❝♦s❛❡❞r♦✳ ✸✸

(47) ✹✳✷✳ ➪❘❊❆❙ ❊ ❱❖▲❯▼❊❙ ❋✐❣✉r❛ ✹✳✶✼✿ ❉✐❛❣♦♥❛❧ d ❞♦ ✐❝♦s❛❡❞r♦✳ P❡❧♦ t❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s t❡♠♦s q✉❡ d2 = (β · α)2 + α2 = α2(β2 + 1). ❙✉❜st✐t✉✐♥❞♦ ♦ ✈❛❧♦r ❞❡ β ❞❛ ❡q✉❛çã♦ ✭✹✳✶✹✮ ✜❝❛♠♦s ❝♦♠  √2  d2 = α2  1+ 2 5 + 1 ❘❡s♦❧✈❡♥❞♦ ♣❛rê♥t❡s❡s✱ ❝♦❝❤❡t❡s ❡ ✐s♦❧❛♥❞♦ d ❝❤❡❣❛♠♦s ❛♦ ✈❛❧♦r ❞❛ ❞✐❛❣♦♥❛❧ ❞♦ ✐❝♦s❛❡❞r♦✳  √ d = α 2(5 + 2 5)  . ❏á s❛❜❡♠♦s q✉❡ ❛s ❛r❡st❛s A ❞❛s ♣✐râ♠✐❞❡s tê♠ ✈❛❧♦r❡s d 2 ✱ ♦✉ s❡❥❛  √ 2(5 + 5) A = α 4 . ❆❣♦r❛ ❝❛❧❝✉❧❛♥❞♦ ♦ ✈❛❧♦r ❞♦ ✈♦❧✉♠❡ ❞❡ ✉♠❛ ❞❛s ✷✵ ♣✐râ♠✐❞❡s q✉❡ ❢♦r♠❛♠ ♦ ✐❝♦s❛❡❞r♦✳ ❈♦♠♦ √❛ ❜❛s❡ é ❢❛❝❡ ❞♦ ✐❝♦s❛❡❞r♦ t❡♠♦s ✉♠ tr✐â♥❣✉❧♦ KLT ❡q✉✐❧át❡r♦ ❞❡ ❧❛❞♦ α ❡ ❛❧t✉r❛ α 2 3✱ ♣♦ré♠ ❛✐♥❞❛ ❢❛❧t❛ ❝❛❧❝✉❧❛r ♦ ✈❛❧♦r ❞❛ ❛❧t✉r❛ ❞❛ ♣✐râ♠✐❞❡✱ ♠❛s ✸✹

(48) ✹✳✷✳ ➪❘❊❆❙ ❊ ❱❖▲❯▼❊❙ √ ♣♦❞❡♠♦s ✉t✐❧✐③❛r ♦ tr✐â♥❣✉❧♦ K GC ✱ ♦♥❞❡ KG é 2 3 ❞❛ ❛❧t✉r❛ ❞♦ t❡tr❛❡❞r♦ ♦✉ α 3 3 ✭✈✐st♦ ♥♦ ❝á❧❝✉❧♦ ❞♦ ✈♦❧✉♠❡ ❞♦ t❡tr❛❡❞r♦✮✳ ❚❡♠♦s q✉❡ CG é ❛ ❛❧t✉r❛ ❞♦ t❡tr❛❡❞r♦ ❋✐❣✉r❛ ✹✳✶✽✿ ❆❧t✉r❛ CG ❞❡ ✉♠❛ ❞❛s ✷✵ ♣✐râ♠✐❞❡s q✉❡ ❢♦r♠❛♠ ♦ ✐❝♦s❛❡❞r♦ ❞❡ ❛r❡st❛ α✳ ❡ CK ❛r❡st❛ ❧❛t❡r❛❧ A ❞❛ ♣✐râ♠✐❞❡✱ ♣♦r ✉♠❛ ❞❛s r❡❧❛çõ❡s ♠étr✐❝❛s ❞♦ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ ♣♦❞❡♠♦s ❛✜r♠❛r q✉❡ 2 CK = 2 CG + 2 KG ⇒ 2 CG = 2 CK − 2 KG s✉❜st✐t✉✐♥❞♦ ♦s ✈❛❧♦r❡s ✜❝❛♠♦s ❝♦♠  √ 2 2 KG = α  2(5 + 4 5)  − √2 α3 3 ❘❡s♦❧✈❡♥❞♦ ❛s ♦♣❡r❛çõ❡s ♥❡❝❡ssár✐❛s ♦❜t❡♠♦s  √ KG = α  6(7 + 3 12 5)  . ❈❛❧❝✉❧❛♥❞♦ ♦ ✈♦❧✉♠❡ ❞❡ ✉♠❛ ♣✐râ♠✐❞❡ ✭✉♠ t❡rç♦ ❞❛ ár❡❛ ❞❛ ❜❛s❡ ✈❡③❡s ❛ ❛❧t✉r❛✮✱ t❡♠♦s √ α2 3 4  ·α· √ 6(7 + 3 5)  12 Vpir = 3 ✸✺

(49) ✹✳✸✳ ❖❯❚❘❖❙ P❖▲■❊❉❘❖❙ r❡s♦❧✈❡♥❞♦ ❛s ♦♣❡r❛çõ❡s ♦❜t❡♠♦s ♦ ✈♦❧✉♠❡ ❞❡ ✉♠❛ ♣✐râ♠✐❞❡ ✈❛❧❡  Vpir = α3  √ 2(7 + 3 5) 48  . ❆ss✐♠ ♦ ✈♦❧✉♠❡ ❞♦ ✐❝♦s❛❡❞r♦ é ✷✵ ✈❡③❡s ♦ ✈♦❧✉♠❡ ❞❡ ✉♠❛ ♣✐râ♠✐❞❡✱ ♣♦rt❛♥t♦ t❡r❡♠♦s  √ 2(7 + 3 5) V = 20 · α3  48   √ (14 + 6 5) = 5 · α3  12  . P♦ré♠ √√ √ 14 + 6 5 = 9 + 6 5 + 5 = (3 + 5)2. P♦rt❛♥t♦ t❡♠♦s q✉❡ ♦ ✈♦❧✉♠❡ ❞♦ ✐❝♦s❛❡❞r♦ ❞❡ ❛r❡st❛ α é √ V = 5 · α3 3+ 5 12 . ✹✳✸ ❖✉tr♦s P♦❧✐❡❞r♦s ❖s ❝❛♣ít✉❧♦s ❡ s❡çõ❡s ❛♥t❡r✐♦r❡s ❛♣r❡s❡♥t❛r❛♠ ❤✐stór✐❛✱ ❞❡✜♥✐çõ❡s✱ ❡❧❡♠❡♥t♦s ❡ r❡❧❛çõ❡s ❡♥tr❡ ❡❧❡s ❡ ♣❧❛♥✐✜❝❛çõ❡s ❞♦s ♣♦❧✐❡❞r♦s ❝♦♥✈❡①♦s r❡❣✉❧❛r❡s✱ ♣♦ré♠ ❛❧❣✉♥s ♣♦❧✐❡❞r♦s ❝♦♥✈❡①♦s ♠❛s ♥ã♦ r❡❣✉❧❛r❡s sã♦ ❜❡♠ ✐♠♣♦rt❛♥t❡s ❡ ❡st✉❞❛❞♦s t❛♥t♦ q✉❛♥t♦ ♦s ♣♦❧✐❡❞r♦s r❡❣✉❧❛r❡s✱ ❛q✉✐ ❢❛r❡♠♦s ✉♠ ❜r❡✈❡ ❝♦♠❡♥tár✐♦ s♦❜r❡ ❡ss❡s ♣♦❧✐❡❞r♦s✳ ❼ P♦❧✐❡❞r♦s ❙❡♠✐rr❡❣✉❧❛r❡s ◆❡st❡ ❝❛♣ít✉❧♦ ❡st✉❞❛♠♦s ♦s ♣♦❧✐❡❞r♦s q✉❡ ♣♦ss✉❡♠ s✉❛s ❢❛❝❡s ❢♦r♠❛❞❛s ❛♣❡♥❛s ♣♦r ✉♠ ú♥✐❝♦ t✐♣♦ ❞❡ ♣♦❧í❣♦♥♦ r❡❣✉❧❛r✱ ♣♦ré♠ ❡①✐st❡♠ ♦✉tr♦s ♣♦❧✐❡❞r♦s q✉❡ sã♦ ❢♦r♠❛❞♦s ♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❝♦♠ ✉♠ ú♥✐❝♦ t✐♣♦✱ ❡ss❡s ♣♦❧✐❡❞r♦s sã♦ ❝❤❛♠❛❞♦s ❞❡ ♣♦❧✐❡❞r♦s s❡♠✐rr❡❣✉❧❛r❡s✳ ❆ ❞❡✜♥✐çã♦ ❛❜❛✐①♦ é ❞❛❞❛ ♣♦r ❏❛♥♦s ❬✶✵❪ ♣á❣✐♥❛ ✽✼✳ ❉❡✜♥✐çã♦ ✹✳✹ ❯♠ ♣♦❧✐❡❞r♦ é ❞✐t♦ s❡♠✐rr❡❣✉❧❛r q✉❛♥❞♦ é ❢♦r♠❛❞♦ ♣♦r ♣♦❧í✲ ❣♦♥♦s r❡❣✉❧❛r❡s✱ ♠❛s ♣♦❞❡ ❡①✐st✐r ♠❛✐s ✉♠ t✐♣♦ ❞❡ ♣♦❧í❣♦♥♦ ❡♠ ❝❛❞❛ ♣♦❧✐❡❞r♦ ✭❣❡r❛❧♠❡♥t❡ ❞♦✐s t✐♣♦s✮✳ ✸✻

(50) ✹✳✸✳ ❖❯❚❘❖❙ P❖▲■❊❉❘❖❙ ❯♠ ❞♦s ❡st✉❞✐♦s♦s ❞❡ss❡ ❣r✉♣♦ ❞❡ ♣♦❧✐❡❞r♦s ❢♦✐ ❆rq✉✐♠❡❞❡s✱ ❡❧❡ ❞❡s❝♦❜r✐✉ ✶✸ ♣♦❧✐❡❞r♦s ❝♦♠ ❡ss❛s ❝❛r❛❝t❡ríst✐❝❛s✱ ❡❧❡s ♣♦❞❡♠ s❡r ♦❜t✐❞♦s r❡❛❧✐③❛♥❞♦ ❝♦rt❡s ♥♦s ♣♦❧✐❡❞r♦s ❞❡ P❧❛tã♦ ❡ ❝♦♥s✐❞❡r❛♥❞♦ q✉❡ ❛s s✉♣❡r❢í❝✐❡s ♦❜t✐❞❛s s❡❥❛♠ ♣♦❧í✲ ❣♦♥♦s r❡❣✉❧❛r❡s✳ ❆❧❣✉♥s ♦❜❥❡t♦s ❛♣r❡s❡♥t❛♠ s❡♠❡❧❤❛♥ç❛ ♦✉ ❛♣❧✐❝❛çõ❡s ❝♦♠♦ ♦s ❞❛ ✜❣✉r❛ ❛❜❛✐①♦✳ ❯♠ ✉t✐❧✐③❛❞♦ ❝♦♠♦ ❛ss❡♥t♦✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♣❛r❛ ❝r✐❛♥ç❛s✱ ♣♦✐s ♦s tr✐â♥❣✉❧♦s ❡q✉✐❧át❡r♦s ❡ ♦❝t❛❡❞r♦s q✉❡ ♦ ❝♦♠♣õ❡♠ q✉❛♥❞♦ ❝♦❧♦r✐❞♦s ❝❤❛♠❛♠ ❛ ❛t❡♥çã♦✱ ❥á ♦ ♦✉tr♦ t❡♠ ❛ s❡♠❡❧❤❛♥ç❛ ✭❥á q✉❡ ✉s❛ ♦s ♠❡s♠♦s ♣♦❧í✲ ❣♦♥♦s ✮ ❝♦♠ ❛ ❜♦❧❛ ✉t✐❧✐③❛❞❛ ♥❛ ❝♦♣❛ ❞❡ ✼✵✳ ❋✐❣✉r❛ ✹✳✶✾✿ P♦❧✐❡❞r♦s ❙❡♠✐rr❡❣✉❧❛r❡s ❼ P✐râ♠✐❞❡s ❉❡✜♥✐çã♦ ✹✳✺ ABCDE❈♦♥s✐❞❡r❡ ✉♠❛ r❡❣✐ã♦ ♣♦❧✐❣♦♥❛❧✱ ♣♦r ❡①❡♠♣❧♦ ✱ ❝♦♥✲ H Vt✐❞❛ ❡♠ ✉♠ ♣❧❛♥♦ ❡ ✉♠ ♣♦♥t♦ ❡①t❡r✐♦r ❛♦ ♣❧❛♥♦ ❞❛ r❡❣✐ã♦ ♣♦❧✐❣♦♥❛❧✳ ❚r❛ç❛♠♦s ♦s s❡❣♠❡♥t♦s V A✱ V B✱ V C ✱ V D ❡ V E✳ ❈❛❞❛ ❞♦✐s ✈ért✐❝❡s ❝♦♥s❡✲ ABCDE V❝✉t✐✈♦s ❞❡ ❞❡t❡r♠✐♥❛♠ ❝♦♠♦ ✉♠❛ r❡❣✐ã♦ tr✐❛♥❣✉❧❛r✳ ❊ss❛ r❡❣✐õ❡s ABC DEtr✐❛♥❣✉❧❛r❡s✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ❛ r❡❣✐ã♦ ♣♦❧✐❣♦♥❛❧ ✱ ❞❡t❡r♠✐♥❛♠ ✉♠ ABC DE V♣♦❧✐❡❞r♦ ❝❤❛♠❛❞♦ ♣✐râ♠✐❞❡ ❞❡ ❜❛s❡ ❡ ✈ért✐❝❡ ✳ ❋✐❣✉r❛ ✹✳✷✵✿ P✐râ♠✐❞❡ ✸✼

(51) ✹✳✸✳ ❖❯❚❘❖❙ P❖▲■❊❉❘❖❙ ❆ ❞❡✜♥✐çã♦ ✹✳✺ ❞❡ ❉❛♥t❡ ❬✻❪ ♣á❣✐♥❛ ✸✼✸ ♥♦s ♠♦str❛ ♦ ❡①❡♠♣❧♦ ❞❡ ✉♠❛ ♣✐râ♠✐❞❡ ❞❡ ❜❛s❡ ♣❡♥t❛❣♦♥❛❧✱ ❛ss✐♠ ❝❧❛ss✐✜❝❛♠♦s ❛s ♣✐râ♠✐❞❡s ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ♣♦❧✐❣♦♥❛❧ ❢♦r♠❛❞❛✱ ✭tr✐❛♥❣✉❧❛r✱ q✉❛❞r❛♥❣✉❧❛r✱ ♣❡♥t❛❣♦♥❛❧✱ ❡ ❛ss✐♠ s✉❝❡ss✐✈❛♠❡♥t❡✮✳ ❆s ♣✐râ♠✐❞❡s ❡①❡r❝❡♠ ❢❛s❝í♥✐♦ s♦❜r❡ ♦ s❡r ❤✉♠❛♥♦ ❞❡s❞❡ ❛ ❆♥t✐❣✉✐❞❛❞❡✱ ❛ ❢♦r♠❛ ♣✐r❛♠✐❞❛❧ t❡♠ r❡ss✉r❣✐❞♦ ♥❛ ❛q✉✐t❡t✉r❛ ♠♦❞❡r♥❛ ❡♠ ❡❞✐❢í❝✐♦s ❞❡ ❣r❛♥❞❡ ✐♠♣♦♥ê♥❝✐❛✳ ❆s ♣✐râ♠✐❞❡s ❞♦ ❊❣✐t♦✱ ❛ ♣✐râ♠✐❞❡ ❞♦ ♠✉s❡✉ ❞♦ ▲♦✉✈r❡ ♦✉ ♠❡s♠♦ ♣✐râ♠✐❞❡s ❞❡❝♦r❛t✐✈❛s✱ sã♦ ❜❡❧♦s ❡①❡♠♣❧♦s ❞❡ss❡ só❧✐❞♦ ♥♦ ♥♦ss♦ ❝♦t✐❞✐❛♥♦✳ ✭✈❡❥❛ ❬✸❪ ♣á❣✐♥❛ ✶✽✸✮ ➪r❡❛ Sb + Sl ❱♦❧✉♠❡ Sb · h 3 ❚❛❜❡❧❛ ✹✳✷✿ ➪r❡❛ ❡ ✈♦❧✉♠❡ ❞❡ ✉♠❛ ♣✐râ♠✐❞❡✳ ❼ Pr✐s♠❛s ❆ ❞❡✜♥✐çã♦ ❛ s❡❣✉✐r ❢♦✐ ❜❛s❡❛❞❛ ♥♦ ❧✐✈r♦ ❞❡ ❉❛♥t❡ ❬✻❪ ♣á❣✐♥❛ ✸✻✹✳ ❉❡✜♥✐çã♦ ✹✳✻ ❈♦♥s✐❞❡r❡ ✉♠❛ r❡❣✐ã♦ ♣♦❧✐❣♦♥❛❧✱ ♣♦r ❡①❡♠♣❧♦ ABCDE✱ ❝♦♥✲ t✐❞❛ ❡♠ ✉♠❛ ♣❧❛♥♦ H✳ ❊s❝♦❧❤❛ ✉♠ ♣♦♥t♦ A′ q✉❛❧q✉❡r ♥ã♦ ♣❡rt❡♥❝❡♥t❡ ❛ H✳ P♦r A′ tr❛❝❡ ♦ ♣❧❛♥♦ J ♣❛r❛❧❡❧♦ ❛ H✳ P❡❧♦s ❞❡♠❛✐s ♣♦♥t♦s B✱ C✱ D✱ E tr❛❝❡ r❡t❛s ♣❛r❛❧❡❧❛s ❛ AA′ q✉❡ ❝♦rt❛♠ J ♥♦s ♣♦♥t♦s B′✱ C′✱ D′✱ E′✳ ❊ss❛s r❡t❛s sã♦ ♣❛r❛❧❡❧❛s ❡♥tr❡ s✐✳ ❋✐❣✉r❛ ✹✳✷✶✿ Pr✐s♠❛ ❚♦♠❡ ❞♦✐s s❡❣♠❡♥t♦s ❝♦♥s❡❝✉t✐✈♦s ❛ss✐♠ ❞❡t❡r♠✐♥❛❞♦s✱ ♣♦r ❡①❡♠♣❧♦ AA′ ❡ BB′✳ ❖ q✉❛❞r✐❧át❡r♦ AA′BB′ é ♣❧❛♥♦✱ ♣♦✐s s❡✉s ❧❛❞♦s AA′ ❡ BB′ sã♦ ♣❛r❛❧❡✲ ❧♦s✳ ■ss♦ ❛❝❛rr❡t❛ q✉❡ AB ❡ A′B′ t❛♠❜é♠ sã♦ ♣❛r❛❧❡❧♦s ✭♣♦✐s ❡stã♦ ❝♦♥t✐❞♦s ❡♠ r❡t❛s ❝♦♣❧❛♥❛r❡s q✉❡ ♥ã♦ s❡ ✐♥t❡rs❡❝t❛♠ ♣♦r ❡st❛r❡♠ ❝♦♥t✐❞❛s ❡♠ ♣❧❛♥♦s ♣❛r❛✲ ❧❡❧♦s✮✳ ▲♦❣♦✱ ♦ q✉❛❞r✐❧át❡r♦ AA′BB′ é ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦✳ ❆s r❡❣✐õ❡s ❧✐♠✐t❛❞❛s ✸✽

(52) ✹✳✸✳ ❖❯❚❘❖❙ P❖▲■❊❉❘❖❙ ♣♦r ♣❛r❛❧❡❧♦❣r❛♠♦s ❛ss✐♠ ❞❡t❡r♠✐♥❛❞♦s✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ❛s r❡❣✐õ❡s ♣♦❧✐❣♦♥❛✐s ABC DE ❡ A′B′C ′D′E′✱ ❞❡t❡r♠✐♥❛♠ ✉♠ ♣♦❧✐❡❞r♦ ❝❤❛♠❛❞♦ ♣r✐s♠❛ ❞❡ ❜❛s❡s ABCDE ❡ A′B′C′D′E′✳ ❆ss✐♠ ❝♦♠♦ ♥❛s ♣✐râ♠✐❞❡s✱ ♦s ♣r✐s♠❛s tê♠ s✉❛s ❝❧❛ss✐✜❝❛çõ❡s ❜❛s❡❛❞❛s ♥❛s ♣♦❧✐❣♦♥❛✐s ❢♦r♠❛❞❛s ♥❛s ❜❛s❡s ✭tr✐❛♥❣✉❧❛r✱ q✉❛❞r❛♥❣✉❧❛r✱ ♣❡♥t❛❣♦♥❛❧✱ ❡ ❛ss✐♠ s✉❝❡ss✐✈❛♠❡♥t❡✮✱ ♥♦ ❡①❡♠♣❧♦ ❛❝✐♠❛ t❡♠♦s ✉♠ ♣r✐s♠❛ ❞❡ ❜❛s❡ ♣❡♥t❛❣♦♥❛❧✳ ❖s ♣r✐s♠❛s sã♦ ♦s ♣♦❧✐❡❞r♦s q✉❡ t❛❧✈❡③ s❡❥❛♠ ✉♠ ❞♦s ♠❛✐s ❢❛❝é✐s ❞❡ ✈✐s✉❛❧✐③❛çã♦ ♥♦ ❞✐❛ ❞✐❛✱ ❞❡s❞❡ ✉♠❛ ❝❛✐①❛ ♣❛r❛ s❛♣❛t♦s✱ ❧❡✐t❡✱ s✉❝♦ ❛ ✉♠ ❝ô♠♦❞♦ ❞❡ ✉♠❛ ❝❛s❛ ♦✉ ✉♠ ♣ré❞✐♦ sã♦ ❡①❡♠♣❧♦s ❞❡ ♣r✐s♠❛s✳ ➪r❡❛ ❱♦❧✉♠❡ 2 · Sb + Sl Sb · h ❚❛❜❡❧❛ ✹✳✸✿ ➪r❡❛ ❡ ✈♦❧✉♠❡ ❞❡ ✉♠ ♣r✐s♠❛✳ ✸✾

(53) ❈❛♣ít✉❧♦ ✺ Pr♦s♣♦st❛s ❞❡ ❙♦❢t✇❛r❡s P♦r ♠❛✐s q✉❡ ❛❧❣✉♠❛s ♣❡ss♦❛s r❡s✐st❛♠ é ✐♥❡❣á✈❡❧ q✉❡ ❛ t❡❝♥♦❧♦❣✐❛ ❝r✐♦✉ ✉♠ ❣r❛♥❞❡ ❧❡q✉❡ ❞❡ ♦♣♦rt✉♥✐❞❛❞❡s ❞❡ ❛♣r❡♥❞✐③❛❣❡♠✳ ◆❛ ♠❛t❡♠át✐❝❛ ❝♦♠♦ ❡♠ q✉❛❧q✉❡r ♦✉tr❛ ár❡❛ ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ♥ã♦ ❢♦✐ ❞✐❢❡r❡♥t❡✱ ♣♦❞❡♠♦s t♦♠❛r ❝♦♠♦ ❡①❡♠♣❧♦ ♠❛✐s s✐♠♣❧❡s ❛ ❝❛❧❝✉❧❛❞♦r❛✱ q✉❡ ❛♥t❡s t✐♥❤❛ ✉s♦ r❡str✐t♦✱ ❤♦ ❥❡ ❞❡ ❛❝❡ss♦ ❢á❝✐❧✱ ♣♦❞❡ s❡r ✉s❛❞❛ ❝♦♠♦ ✉♠❛ ❣r❛♥❞❡ ❢❡rr❛♠❡♥t❛ ♣❛r❛ ❞✐❝❡♥t❡s ❡ ❞♦❝❡♥t❡s✳ ❆ ♣r♦♣♦st❛ ❞❡ss❡ ❝❛♣ít✉❧♦ é ❛♣r❡s❡♥t❛r ❛❧❣✉♠❛s ❞❡ss❛s ♥♦✈❛s ❢❡rr❛♠❡♥t❛s q✉❡ ♣♦❞❡♠ s❡r ✉t✐❧✐③❛❞❛s ♥❛ ❛♣r❡❞✐③❛❣❡♠ ❞♦ ❛❧✉♥♦✳ ✺✳✶ ❈❛❜r✐ ❈❛❜r✐✲●❡♦♠❡tr② é ✉♠ s♦❢t✇❛r❡ ❞❡ ❝♦♥str✉çã♦ ❡♠ ❣❡♦♠❡tr✐❛ ❞❡s❡♥✈♦❧✈✐❞♦ ♣❡❧♦ ■♥st✐t✉t ❞✬■♥❢♦r♠❛t✐q❡ ❡t ❞❡ ▼❛t❤❡♠❛t✐q✉❡s ❆♣♣❧✐q✉❡❡s ❡♠ ●r❡♥♦❜❧❡ ✭■▼❆●✮ ❡ é ♦ r❡s✉❧t❛❞♦ ❞❛ ❝♦❧❛❜♦r❛çã♦ ❝♦♥st❛♥t❡ ❞❡ ❝✐❡♥t✐st❛s ❞❛ ✐♥❢♦r♠át✐❝❛✱ ❡s♣❡❝✐❛❧✐st❛s ❡♠ ❡❞✉❝❛çã♦ ❡ ♣r♦❢❡ss♦r❡s✳ ❙♦❢t✇❛r❡ ❈❛❜r✐ ❢♦✐ ❞✐str✐❜✉í❞♦ ❡♠ t♦❞♦ ♦ ♠✉♥❞♦ ❤á ♠❛✐s ❞❡ ✷✵ ❛♥♦s✳ ❈♦♥❤❡❝✐❞♦ ♣♦r s❡✉ ❢♦❝♦ ♣❡❞❛❣ó❣✐❝♦✱ ♦s ♣r♦❞✉t♦s ❞❡ s♦❢t✇❛r❡ ❈❛❜r✐ ❥á sã♦ ✉s❛❞♦s ♣♦r ♠❛✐s ❞❡ ✶✵✵ ♠✐❧❤õ❡s ❞❡ ✉s✉ár✐♦s✳ ❉❡s❡♥✈♦❧✈✐❞♦ ♣❛r❛ ♦ ❡♥s✐♥♦ ♠é❞✐♦ ❡ ♦s ❛❧✉♥♦s ❞♦ ❡♥s✐♥♦ ♠é❞✐♦✱ ❡st❡ s♦❢t✇❛r❡ ♣❡r♠✐t❡ q✉❡ ♦s ❛❧✉♥♦s ❛ ♠❛♥✐♣✉❧❛r ❞✐r❡t❛♠❡♥t❡ ♦s ♦❜❥❡t♦s ♠❛t❡♠át✐❝♦s ✭á❧❣❡❜r❛✱ ❛♥á❧✐s❡✱ ❣❡♦♠❡tr✐❛✱ tr✐❣♦♥♦♠❡tr✐❛✳✳✳✮ ♦✉ ♦❜❥❡t♦s ❢ís✐❝♦s ✭♠❡❝â♥✐❝❛✱ ó♣t✐❝❛✳✮ P❛r❛ q✉❡ ♦s ❛❧✉♥♦s ♣♦ss❛♠ ♠❛✐s ❢❛❝✐❧♠❡♥t❡ ❝♦♠♣r❡❡♥❞❡r ♦s ❝♦♥❝❡✐t♦s✳ ❆s ✐♥❢♦r♠❛çõ❡s ❞♦ s♦❢t✇❛r❡ ❡stã♦ ❞✐s♣♦♥í✈❡✐s ♥♦ ❡♥❞❡r❡ç♦ ❡❧❡trô♥✐❝♦ ❤tt♣✿✇✇✇✳❝❛❜r✐✳❝♦♠ ❬✶✼❪✱ ❧á ❡♥❝♦♥tr❛♠♦s ✈❡rsõ❡s ❞✐s♣♦♥í✈❡✐s ♣❛r❛ ❞♦✇♥❧♦❛❞✱ tr❡✐✲ ♥❛♠❡♥t♦ ♦♥✲❧✐♥❡✱ ❛♣♦✐♦ ❡♥tr❡ ♦✉tr❛s ✐♥❢♦r♠❛çõ❡s q✉❡ ❛❥✉❞❛♠ ♥❛ ✐♥st❛❧❛çã♦ ❡ ✉s♦ ❞♦ s♦❢t❛✇❛r❡✳ ❊①✐st❡♠ ✈ár✐❛s ✈❡rsõ❡s ❞♦ ❝❛❜r✐ ❛q✉✐ ✉t✐❧✐③❛♠♦s ♦ ❝❛❜r✐ ✸❉✱ ♥❡❧❡ ♣♦❞❡♠♦s ❝♦♥s✲ tr✉✐r ✈ár✐♦s ♣♦❧✐❡❞r♦s ❝♦♥✈❡①♦s✱ ♥❛ ❜❛rr❛ ❞❡ ❢❡rr❛♠❡♥t❛s ❡♥❝♦♥tr❛♠♦s ❛❧❣✉♠❛s ❥❛♥❡✲ ❧❛s✱ ♥❡❧❛s t❡r❡♠♦s t♦❞❛ ❛s ✐♥❢♦r♠❛çõ❡s ♣❛r❛ ❝♦♥str✉ír♠♦s ✉♠ ♣♦❧✐❡❞r♦✳ ✹✵

(54) ✺✳✷✳ P❖▲❨ ❋✐❣✉r❛ ✺✳✶✿ ❏❛♥❡❧❛s ❞♦ ❈❛❜r✐ ✸❉✳ ◆❛ ✜❣✉r❛ ♦❜s❡r✈❛♠♦s q✉❡ ❛ ❥❛♥❡❧❛ s❡❧❡❝✐♦♥❛❞❛ ❢♦✐ ❛ ❞♦s ♣♦❧✐❡❞r♦s r❡❣✉❧❛r❡s✱ ♥❡❧❛ ❡♥❝♦♥tr❛♠♦s t♦❞♦s ♦s ❝✐♥❝♦ ♣♦❧✐❡❞r♦s✱ s❡❧❡❝✐♦♥❛♥❞♦ ❛❧❣✉♠ ❞❡❧❡s ♣♦❞❡♠♦s ♦❜s❡r✈❛r s❡✉s ✈ért✐❝❡s✱ ❛r❡st❛s✱ ❢❛❝❡s✱ ♣❧❛♥✐✜❝❛çõ❡s ❡♥tr❡ ♦✉tr❛s ✐♥❢♦r♠❛çõ❡s q✉❡ ❢❛❝✐❧✐t❛♠ ❛ ❝♦♠♣r❡❡♥sã♦ ❞♦s ❡❧❡♠❡♥t♦s ❡ r❡❧❛çõ❡s ❡♥❝♦♥tr❛❞❛s ♥♦s ♣♦❧✐❡❞r♦s✳ ◆❛ ✜❣✉r❛ ✺✳✷ ✈❡♠♦s ✉♠ ❤❡①❛❡❞r♦ ✭❝✉❜♦✮✱ ♣♦❞❡♠♦s ♣❧❛♥✐✜❝á✲❧♦✱ r♦t❛❝✐♦♥á✲❧♦ ❡♥tr❡ ♦✉tr❛s ♦♣çõ❡s✱ é ✐♥t❡r❡ss❛♥t❡ q✉❡ ♦ ❛❧✉♥♦ ♣♦ss❛ ♠❛♥✉s❡❛r ✭♣r✐♥❝✐♣❛❧♠❡♥t❡ ♦s q✉❡ tê♠ ♠❛✐♦r ❞✐✜❝✉❧❞❛❞❡✮ ❝❛s♦ ❛♣❡♥❛s ♦ ♣r♦❢❡ss♦r t✐✈❡r ❛❝❡ss♦ ❛♦ s♦❢t✇❛r❡ ❞❡ ❢♦r♠❛ ❛ ♣❡r❝❡❜❡r ❝❛r❛❝t❡ríst✐❝❛s ♥❛ ❢♦r♠❛çã♦ ❞♦ ♣♦❧✐❡❞r♦✳ ❋✐❣✉r❛ ✺✳✷✿ P❧❛♥✐✜❝❛♥❞♦ ♦ ❝✉❜♦✳ ✺✳✷ P♦❧② ❖✉tr♦ s♦❢t✇❛r❡ ❝❛♣❛③ ❞❡ ♠♦str❛r ♦s ♣♦❧✐❡❞r♦s ❞❡ ♠❛♥❡✐r❛ ❜❡♠ ❞✐♥â♠✐❝❛ é ♦ P♦❧②✳ ❆s ✐♥❢♦r♠❛çõ❡s s♦❜r❡ ♦ s♦❢t❛✇❛r❡ ❡stã♦ ❞✐s♣♦♥í✈❡✐s ❡♠ ❤tt♣✿✴✴✇✇✇✳♣❡❞❛✳❝♦♠✴♣♦❧② ✹✶

(55) ✺✳✸✳ ❖❯❚❘❖❙ ❙❖❋❚❲❆❘❊❙ ❬✶✽❪ ♥❡❧❡ ❡♥❝♦♥tr❛♠♦s ❝♦♠♦ ❢❛③❡r ❞♦✇♥❧♦❛❞ ❞♦ s♦❢t✇❛r❡✱ ✐♥❢♦r♠❛çõ❡s s♦❜r❡ ❛ ❡♠♣r❡s❛ ❡♥tr❡ ♦✉tr♦s ♣r♦❞✉t♦s ♦❢❡r❡❝✐❞♦s✳ P♦❧② é ✉♠ ♣r♦❣r❛♠❛ s❤❛r❡✇❛r❡ ✭q✉❡ ❡stá ❞✐s♣♦♥í✈❡❧ ❞❡ ❢♦r♠❛ ❣r❛t✉✐t❛ ❡✱ ♠✉✐t❛s ✈❡③❡s ✐♥❢♦r♠❛❧♠❡♥t❡ ❞✐str✐❜✉í❞♦ ♣❛r❛ ❛✈❛❧✐❛çã♦✱ ❛♣ós ♦ q✉❛❧ ❛ t❛①❛ ♣♦❞❡ s❡r s♦❧✐❝✐✲ t❛❞❛ ♣❛r❛ ✉s♦ ❝♦♥t✐♥✉❛❞♦✮✱ ♣❛r❛ ❡①♣❧♦r❛çã♦ ❡ ❝♦♥str✉çã♦ ❞❡ ♣♦❧✐❡❞r♦s ✳ ❈♦♠ P♦❧②✱ é ♣♦ssí✈❡❧ ♠❛♥✐♣✉❧❛r ♦s só❧✐❞♦s ♣♦❧✐é❞r✐❝♦s ♥♦ ❝♦♠♣✉t❛❞♦r ❡♠ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞❡ ❢♦r✲ ♠❛s✳ ❆ ♠❛♥✐♣✉❧❛çã♦ ❞♦ P♦❧② é ❜❡♠ ❢á❝✐❧ ♥❡❧❡ t❡♠♦s ✉♠❛ ❜❛rr❛ ❞❡ ❢❡rr❛♠❡♥t❛s ❝♦♠ ♣♦✉❝❛s ♦♣çõ❡s✱ ♠❛s ❝♦♠ ♠✉✐t❛s ♣♦ss✐❜✐❧✐❞❛❞❡s ♣❛r❛ ✈✐s✉❛❧✐③❛çõ❡s✳ ❋✐❣✉r❛ ✺✳✸✿ ❇❛rr❛ ❞❡ ❢❡rr❛♠❡♥t❛s ❞♦ P♦❧②✳ ◆❡❧❛ t❡♠♦s t♦❞♦s ♦s ♣♦❧í❣♦♥♦s r❡❣✉❧❛r❡s✱ s❡♠✐rr❡❣✉❧❛r❡s✱ ♣r✐s♠❛s ❡ ❛♥t✐✲♣r✐♠❛s ❡♥tr❡ ♦✉tr♦s ♣♦❧✐❡❞r♦s✱ ♦♣çõ❡s ❞❡ ♣❧❛♥✐✜❝❛çõ❡s ❝♦♠♦ ❞❛ ✜❣✉r❛ ❛❜❛✐①♦✳ ❋✐❣✉r❛ ✺✳✹✿ P❧❛♥✐✜❝❛♥❞♦ ♦ ✐❝♦s❛❡❞r♦✳ ✺✳✸ ❖✉tr♦s ❙♦❢t✇❛r❡s ❆❧é♠ ❞♦ ❈❛❜r✐ ❡ P♦❧② ✭❡ s✉❛s ✈❡rsõ❡s✮ t❡♠♦s ♦✉tr♦s s♦❢t✇❛r❡s ❝❛♣❛③❡s ❞❡ ❛✉①✐❧✐❛r ♥♦ tr❛❜❛❧❤♦ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛ t❛♥t♦ ♥❛ ●❡♦♠❡tr✐❛ P❧❛♥❛ ❡ ❊s♣❛❝✐❛❧ ❛s ✐♥❢♦r♠❛çõ❡s ❢♦r❛♠ ❝♦❧❤✐❞❛s ♥♦ s✐t❡ ❞❛ ❯❋❘●❙ ❬✼❪✱ ♥❡❧❡ t❡♠♦s ❝♦♠♦ r❡❛❧✐③❛r ♦ ❞♦✇♥❧♦❛❞ ❞♦s s♦❢t✇❛r❡s ❡♥tr❡ ♦✉tr❛s ✐♥❢♦r♠❛çõ❡s✳ ❆❧❣✉♥s ❡①❡♠♣❧♦s✿ ✹✷

(56) ✺✳✸✳ ❖❯❚❘❖❙ ❙❖❋❚❲❆❘❊❙ ❼ ❈■◆❉❊❘❊▲❆ ✲ ❙♦❢t✇❛r❡ ❞❡ ❝♦♥str✉çã♦ ❡♠ ❣❡♦♠❡tr✐❛ ❞❡s❡♥✈♦❧✈✐❞♦ ♣♦r ❏ür❣❡♥ ❘✐❝❤t❡r✲●❡❜❡rt ✫ ❯❧r✐❝❤ ❑♦rt❡♥❦❛♠♣ ❝♦♠❡r❝✐❛❧✐③❛❞♦ ♣♦r ❙✉♥ ▼✐❝r♦s②st❡♠s✱ ■♥❝✳ ➱ ✉♠ s♦❢t✇❛r❡ ❞❡ ❝♦♥str✉çã♦ q✉❡ ♥♦s ♦❢❡r❡❝❡ ✧ré❣✉❛ ❡ ❝♦♠♣❛ss♦ ❡❧❡trô✲ ♥✐❝♦s✧✱ s❡♠❡❧❤❛♥t❡ ❛♦ ❈❛❜r✐✳ ❯♠ ❞✐❢❡r❡♥❝✐❛❧ ❞❡st❡ s♦❢t✇❛r❡ é q✉❡ ♣❡r♠✐t❡ q✉❡ s❡ tr❛❜❛❧❤❡ t❛♠❜é♠ ❡♠ ❣❡♦♠❡tr✐❛ ❤✐♣❡r❜ó❧✐❝❛ ❡ ❡s❢ér✐❝❛✳ ❊ ♠❛✐s✿ t❡♠ ❛ ♦♣çã♦ ❞❡ s❛❧✈❛r ❝♦♠♦ ♣á❣✐♥❛ ❞❛ ✇❡❜ ❛✉t♦♠❛t✐❝❛♠❡♥t❡✳ ❼ ❉❘ ●❊❖ ✲ ❙♦❢t✇❛r❡ ❞❡ ❝♦♥str✉çã♦ ❡♠ ❣❡♦♠❡tr✐❛ ❞❡s❡♥✈♦❧✈✐❞♦ ♣♦r ❍✐❧❛✐r❡ ❋❡r♥❛♥❞❡ ●r❡♥♦❜❧❡ ❡ q✉❡ ♥♦s ♦❢❡r❡❝❡ ✧ré❣✉❛ ❡ ❝♦♠♣❛ss♦ ❡❧❡trô♥✐❝♦s✧✱ s❡♥❞♦ ❛ ✐♥t❡r❢❛❝❡ ❞❡ ♠❡♥✉s ❞❡ ❝♦♥str✉çã♦ ❡♠ ❧✐♥❣✉❛❣❡♠ ❝❧áss✐❝❛ ❞❛ ●❡♦♠❡tr✐❛✳ ❖s ❞❡s❡♥❤♦s ❞❡ ♦❜❥❡t♦s ❣❡♦♠étr✐❝♦s sã♦ ❢❡✐t♦s ❛ ♣❛rt✐r ❞❛s ♣r♦♣r✐❡❞❛❞❡s q✉❡ ♦s ❞❡✜♥❡♠ ❡ ♠❛♥tê♠ ❡st❛❜✐❧✐❞❛❞❡ s♦❜ ♦ ♠♦✈✐♠❡♥t♦✳ ❼ ●❊❖❙P❆❈❊ ✲ ❙♦❢t✇❛r❡ ❞❡ ❝♦♥str✉çã♦ ❡ ❡①♣❧♦r❛çã♦ ❡♠ ❣❡♦♠❡tr✐❛ q✉❡ tr❛❜❛❧❤❛ ♦s ❝♦♥❝❡✐t♦s ❡s♣❛❝✐❛✐s✳ ❉❡s❡♥✈♦❧✈✐❞♦ ♣❡❧♦ ❈❡♥tr❡ ❞❡ ❘❡❝❤❡r❝❤❡ ❡t ❞✬❊①♣ér✐♠❡♥t❛t✐♦♥ ♣♦✉r ❧✬❊♥s✐❣♥❡♠❡♥t ❞❡s ▼❛t❤é♠❛t✐q✉❡s ❈❘❊❊▼✳ ❼ ●❘❊❆❚ ❙❚❊▲▲❆ ✲ ❙♦❢t✇❛r❡ q✉❡ tr❛❜❛❧❤❛ ❝♦♠ ❛ ✈✐s✉❛❧✐③❛çã♦✱ r♦t❛çã♦ ❡ ❝♦♥s✲ tr✉çã♦ ❞❡ ♣♦❧✐❡❞r♦s ❝♦♥✈❡①♦s ❡ ♥ã♦ ❝♦♥✈❡①♦s✳ ❼ ❙❑❊❚❈❍P❆❉ ✲ ❙♦❢t✇❛r❡ ❞❡ ❝♦♥str✉çã♦ ❡♠ ❣❡♦♠❡tr✐❛ ❞❡s❡♥✈♦❧✈✐❞♦ ♣♦r ◆✳ ❏❛❝❦✐✇ ❡ ❙✳❙t❡❦❡t❡❡ ❝♦♠❡r❝✐❛❧✐③❛❞♦ ♣♦r ❑❡② ❈✉rr✐❝✉❧✉♠ Pr❡ss✱ ♥♦s ♦❢❡r❡❝❡ ✧ré❣✉❛ ❡ ❝♦♠♣❛ss♦ ❡❧❡trô♥✐❝♦s✧✱ s❡♥❞♦ ❛ ✐♥t❡r❢❛❝❡ ❞❡ ♠❡♥✉s ❞❡ ❝♦♥str✉çã♦ ❡♠ ❧✐♥❣✉❛❣❡♠ ❝❧áss✐❝❛ ❞❛ ●❡♦♠❡tr✐❛✳ ❖s ❞❡s❡♥❤♦s ❞❡ ♦❜❥❡t♦s ❣❡♦♠étr✐❝♦s sã♦ ❢❡✐t♦s ❛ ♣❛rt✐r ❞❛s ♣r♦♣r✐❡❞❛❞❡s q✉❡ ♦s ❞❡✜♥❡♠ ❡ ♠❛♥tê♠ ❡st❛❜✐❧✐❞❛❞❡ s♦❜ ♦ ♠♦✈✐♠❡♥t♦✳ ➱ ♣♦ssí✈❡❧ ❝♦♥✈❡rt❡r s❡✉s ❛rq✉✐✈♦s ❡♠ ❧✐♥❣✉❛❣❡♠ ❏❆❱❆✱ ❞❡ ♠❛♥❡✐r❛ q✉❡ s❡❥❛♠ ❞✐s♣♦♥✐❜✐❧✐③❛❞♦s ♥❛ r❡❞❡✳ ❼ ❲■◆●❊❖▼ ✲ ❙♦❢t✇❛r❡ q✉❡ ♣❡r♠✐t❡ ❝♦♥str✉çõ❡s ❣❡♦♠étr✐❝❛s ❜✐❞✐♠❡♥s✐♦♥❛✐s ❡ tr✐❞✐♠❡♥s✐♦♥❛✐s✳ ❚r❛③❡♠♦s ❛q✉✐ ❛❧❣✉♠❛s q✉❡stõ❡s q✉❡ ♣♦❞❡♠ s❡r✈✐r ♣❛r❛ ✈❡r✐✜❝❛çã♦ ❞❡ ❛♣r❡♥❞✐✲ ③❛❣❡♠ ❞♦ ❛❧✉♥♦✱ ❝❧❛r♦ q✉❡ ♦ ♣r♦❢❡ss♦r ♣♦❞❡ ❛❞❡q✉á✲❧❛ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ s✉❛ t✉r♠❛✱ ❛❧❣✉♠❛s ❞❛s q✉❡stõ❡s ♣♦❞❡♠ s❡r r❡s♦❧✈✐❞❛s ❝♦♠ ♦ ✉s♦ ❞❡ s♦❢t✇❛r❡ ❡ ❝❧❛r♦ q✉❡ s❡ ♦ ❛❧✉♥♦ t✐✈❡r ❛❝❡ss♦ ❛ ✉♠ ❞❡❧❡s ❞❡✐①❛r q✉❡ ❡❧❡ ♠❛♥✉s❡✐✳ ✶✳ ❉❡✜♥❛ P♦❧✐❡❞r♦s ✷✳ ❈♦♠ ❛✉①í❧✐♦ ❞❡ ✉♠❛ ré❣✉❛ ❞❡s❡♥❤❡ ✉♠ ♣♦❧í❣♦♥♦ ❝♦♥✈❡①♦ ❡ ♦✉tr♦ ♥ã♦ ❝♦♥✈❡①♦ q✉❡ t❡♥❤❛♠ ❛ ♠❡s♠❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❧❛❞♦s✳ ✸✳ ❈✐t❡ ♦s ♥♦♠❡s ❞♦s ❝✐♥❝♦ P♦❧✐❡❞r♦s ❘❡❣✉❧❛r❡s ✹✳ ❈♦♠♣❧❡t❡ ❛ t❛❜❡❧❛✿ ✹✸

(57) ✺✳✸✳ ❖❯❚❘❖❙ ❙❖❋❚❲❆❘❊❙ P♦❧✐❡❞r♦ ❚✐♣♦ ❞❡ ❋❛❝❡ ◆ú♠❡r♦ ❞❡ ❛r❡st❛s q✉❡ ✐♥❝✐❞❡♠ ❡♠ ✉♠ ✈ért✐❝❡ ❚❡tr❛❡❞r♦ ❍❡①❛❡❞r♦ ❖❝t❛❡❞r♦ ❉♦❞❡❝❛❡❞r♦ ■❝♦s❛❡❞r♦ ✺✳ P❧❛♥✐✜q✉❡ ❞❡ ❞♦✐s ♠♦❞♦s ❞✐❢❡r❡♥t❡s ✉♠ ❝✉❜♦✳ ✻✳ ❯♠❛ ♣❡ss♦❛ q✉❡ ♦❜s❡r✈❛ ✉♠ ♦❜❥❡t♦ q✉❡ ❡stá ♥✉♠❛ s❛❧❛ ❡ ✈❡r✐✜❝❛ q✉❡ ❡ss❡ é ✉♠ ♣♦❧✐❡❞r♦ ❝♦♥✈❡①♦ ❡ q✉❡ t♦❞❛s ❛s ❢❛❝❡s sã♦ tr✐❛♥❣✉❧❛r❡s ❡ q✉❡ ❡♠ ❝❛❞❛ ✈ért✐❝❡ ✐♥❝✐❞❡♠ q✉❛tr♦ ❛r❡st❛s✱ ❡ss❡ ♦❜❥❡t♦ t❡♠ ❛ ❢♦r♠❛ ❞❡ ✉♠✿ ❛✮ ♦❝t❛❡❞r♦ ❜✮ ♣r✐s♠❛ ❞❡ ❜❛s❡ ♣❡♥t❛❣♦♥❛❧ ❝✮ ❞♦❞❡❝❛❡❞r♦ ❞✮ ♣r✐s♠❛ ❞❡ ❜❛s❡ ♦❝t♦❣♦♥❛❧ ❡✮ ✐❝♦s❛❡❞r♦ ✼✳ ❙❛❜❡♥❞♦ q✉❡ ✉♠ ♣♦❧✐❡❞r♦ ♣♦ss✉✐ ❞✉❛s ❢❛❝❡s ♣❡♥t❛❣♦♥❛✐s ❡ ❝✐♥❝♦ ❢❛❝❡s r❡t❛♥❣✉✲ ❧❛r❡s✱ ❞❡t❡r♠✐♥❡ q✉❛♥t♦s ✈ért✐❝❡s ♣♦ss✉❡ ❡ss❡ ♣♦❧✐❡❞r♦✳ ✽✳ ❯♠ ♣♦❧✐❡❞r♦ ❝♦♥✈❡①♦ ♣♦ss✉❡ s❡✐s ❛r❡st❛s ❛ ♠❛✐s q✉❡ ♦ ♥ú♠❡r♦ ❞❡ ✈ért✐❝❡s✱ q✉❛♥t♦s ✈ért✐❝❡s ❡ss❡ ♣♦❧✐❡❞r♦ ♣♦ss✉❡ s❛❜❡♥❞♦ q✉❡ ❡❧❡ t❡♠ ✹ ❢❛❝❡s ❤❡①❛❣♦♥❛✐s ❡ ✹ ❢❛❝❡s tr✐❛♥❣✉❧❛r❡s✳ ✾✳ ❖❜s❡r✈❡ ❛ ♣❧❛♥✐✜❝❛çã♦ ❛❜❛✐①♦✿ ❆♥❛❧✐s❛♥❞♦ ❛ ♣❧❛♥✐✜❝❛çã♦ ♣♦❞❡♠♦s ❛✜r♠❛r q✉❡ tr❛t❛✲s❡ ❞❡✿ ❛✮ ✉♠❛ ♣✐râ♠✐❞❡ ❞❡ ❜❛s❡ tr✐❛♥❣✉❧❛r ❜✮ ✉♠ ♣r✐s♠❛ ❞❡ ❜❛s❡ tr✐❛♥❣✉❧❛r ❝✮ ✉♠❛ ♣✐râ♠✐❞❡ ❞❡ ❜❛s❡ q✉❛❞r❛♥❣✉❧❛r ✹✹

(58) ✺✳✸✳ ❖❯❚❘❖❙ ❙❖❋❚❲❆❘❊❙ ❞✮ ✉♠ ♣r✐s♠❛ ❞❡ ❜❛s❡ q✉❛❞r❛♥❣✉❧❛r ❡✮ ✉♠ ♣♦❧✐❡❞r♦ r❡❣✉❧❛r ✶✵✳ ❉❡t❡r♠✐♥❡ ❛ ár❡❛ ❞❛ s✉♣❡r❢í❝✐❡ ❞❡ ✉♠ ♦❝t❛❡❞r♦ ❡ ✉♠ ✐❝♦s❛❡❞r♦ ❞❡ ❛r❡st❛s ✺ ❝♠ ❡ ✷✱✷✺ ❝♠ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ✶✶✳ ❈❛❧❝✉❧❡ ❛ ár❡❛ ❞❛ s✉♣❡r❢í❝✐❡ ❡ ♦ ✈♦❧✉♠❡ ❞❡ ✉♠❛ ♣✐râ♠✐❞❡ q✉❡ t❡♠ ❝♦♠♦ ❜❛s❡ ✉♠ tr✐â♥❣✉❧♦ ❡q✉✐❧át❡r♦ ❞❡ ♠❡❞✐❞❛ ✽ ❝♠ ❡ ❛r❡st❛s ❧❛t❡r❛✐s ❝♦♠ ✶✵ ❝♠✳ ✶✷ ❯♠❛ ❝❛✐①❛ ❞á❣✉❛ t❡♠ ❢♦r♠❛ ❞❡ ✉♠ ❝✉❜♦ ❞❡ ❛r❡st❛ ✷✱✺ ♠ ❝❛❧❝✉❧❡ ❛ ❝❛♣❛❝✐❞❛❞❡ ♠á①✐♠❛✱ ❡♠ ❧✐tr♦s✱ ❞❡ ❛r♠❛③❡♥❛♠❡♥t♦ ❞❡ss❛ ❝❛✐①❛✳ ✹✺

(59) ❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ❬✶❪ ➪❱■▲❆✱ ●❡r❛❧❞♦✳ ❊✉❝❧✐❞❡s✱ ❣❡♦♠❡tr✐❛ ❡ ❢✉♥❞❛♠❡♥t♦s✳ ❘❡✈✐st❛ Pr♦❢❡ss♦r ❞❡ ▼❛✲ ➏t❡♠át✐❝❛✱ ❙ã♦ P❛✉❧♦✳ ❙♦❝✐❡❞❛❞❡ ❇r❛s✐❧❡✐r❛ ❞❡ ▼❛t❡♠át✐❝❛✱ ✹✺✱ ✷✵✵✶ ❬✷❪ ❆❩❆▼❇❯❏❆ ❋■▲❍❖✱ ❩✳ ❉❡♠♦♥str❛çã♦ ❞♦ t❡♦r❡♠❛ ❞❡ ❊✉❧❡r ♣❛r❛ ♣♦❧✐❡❞r♦s ➏❝♦♥✈❡①♦s✳ ❘❡✈✐st❛ ❞♦ Pr♦❢❡ss♦r ❞❡ ▼❛t❡♠át✐❝❛✱ ❙ã♦ P❛✉❧♦✳ ❙♦❝✐❡❞❛❞❡ ❇r❛s✐❧❡✐r❛ ❞❡ ▼❛t❡♠át✐❝❛✱ ✸✱ ♣✳ ✶✺✲✶✼✱ ✶✾✽✸✳ ❬✸❪ ❇❆❘❘❖❙❖✱ ❏✉❧❧✐❛♥❡ ▼❛ts✉❜❛r❛✱ ❈♦♥❡①õ❡s ❝♦♠ ❛ ▼❛t❡♠át✐❝❛✳ ❙ã♦ P❛✉❧♦✳ ✶✳❡❞✳✱ ▼♦❞❡r♥❛✱ ✷✵✶✵✳ ✹✹✵♣✳ ❬✹❪ ❇■❆◆❈❍■◆■✱ ❊❞✇❛❧❞♦❀ P❆❈❈❖▲❆✱ ❍❡r✈❛❧❀ ▼❛t❡♠át✐❝❛ ✷✳ ❙ã♦ P❛✉❧♦✳ ✷✳❡❞✳✱ ▼♦❞❡r♥❛✱ ✶✾✾✺✳ ✸✹✻♣✳ ❬✺❪ ❇❖❨❊❘✱ ❈✳ ❇✳✱❍✐stór✐❛ ❞❛ ▼❛t❡♠át✐❝❛✳ ❙P✱ ❊❞❣❛r ❇❧✉❝❤❡r ▲t❞❛✳ ❚r❛❞✉çã♦ ❞❡ ❊❧③❛ ❋✳ ●♦♠✐❞❡✳ ✶✾✼✹✳ ✹✽✽♣✳ ❬✻❪ ❉❆◆❚❊✱ ▲✉✐③ ❘♦❜❡rt♦✱ ▼❛t❡♠át✐❝❛✱ ✈♦❧✉♠❡ ú♥✐❝♦✱ ❙ã♦ P❛✉❧♦✳ ✶✳ ❡❞✳✱➪t✐❝❛✱ ✷✵✵✺✳ ✺✵✹♣✳ ❬✼❪ ❊❉❯❈❆❈➹❖ ▼❆❚❊▼➪❚■❈❆ ❊ ❚❊❈◆❖▲❖●■❆ ■◆❋❖❘▼➪❚■❈❆✱ ❉✐s♣♦♥í✲ ✈❡❧ ❡♠✿ ❤tt♣✿✴✴✇✇✇✷✳♠❛t✳✉❢r❣s✳❜r✴❡❞✉♠❛t❡❝✴s♦❢t✇❛r❡s✴s♦❢t❴❣❡♦♠❡tr✐❛✳♣❤♣✳ ❆❝❡ss♦ ❡♠✿ ✶✸ ❥✉❧✳ ✷✵✶✹ ❬✽❪ ●❘❆◆❏❆✱ ❈✳❊✳❙✳❈✳❀ ❈❖❙❚❆✱ ▼✳P✳▼✳✳ ❆ ❋ór♠✉❧❛ ❞♦ ❱♦❧✉♠❡ ❞♦ ■❝♦s❛❡❞r♦ ➏❘❡✈✐st❛ ❞♦ Pr♦❢❡ss♦r ❞❡ ▼❛t❡♠át✐❝❛✱ ❙ã♦ P❛✉❧♦✳ ❙♦❝✐❡❞❛❞❡ ❇r❛s✐❧❡✐r❛ ❞❡ ▼❛t❡✲ ♠át✐❝❛✱ ❛♣♦✐♦ ❯❙P✳ ✼✹✱ ✷✵✶✶✳ ❬✾❪ ■❊❩❩■✱ ●❡❧s♦♥✱ ❡t✳❛❧✳✱ ▼❛t❡♠át✐❝❛✱ ✈♦❧✉♠❡ ú♥✐❝♦✳ ❙ã♦ P❛✉❧♦✳ ❆t✉❛❧✱ ✷✵✵✷✳ ✻✺✽♣✳ ❬✶✵❪ ❏❆◆❖❙✱ ▼✐❝❤❡❧✱ ▼❛t❡♠át✐❝❛ ♣❛r❛ ♣❛✐s ❡ ✐♥t❡r❡ss❛❞♦s✱ ✈♦❧✉♠❡ ✷✱ ❙ã♦ P❛✉❧♦✳ ✶✳❡❞✳✱ ▲✐✈r❛r✐❛ ❞❛ ❋ís✐❝❛✱ ✷✵✶✶✳ ✹✹✵♣✳ ❬✶✶❪ ▲■▼❆✱ ❊❧♦♥ ▲❛❣❡s✱ ▼❡✉ Pr♦❢❡ss♦r ❞❡ ▼❛t❡♠át✐❝❛ ❡ ♦✉tr❛s ❤✐stór✐❛s✱ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✳ ■▼P❆✳ ✶✾✾✶✳ ✹✻

(60) ❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá❢✐❝❛s ❬✶✷❪ ▲■▼❆✱ ❊❧♦♥ ▲❛❣❡s✱ ❡t✳❛❧✳✱ ❆ ▼❛t❡♠át✐❝❛ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ✈♦❧✉♠❡ ✷✱ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✳ ✻✳❡❞✳✱ ❙❇▼✱ ✷✵✵✻✳ ✸✼✸♣✳ ❬✶✸❪ P❆■❱❆✱ ▼❛♥♦❡❧✱ ▼❛t❡♠át✐❝❛✱ ✈♦❧✉♠❡ ú♥✐❝♦✱ ❙á♦ P❛✉❧♦✳ ✶✳❡❞✳✱ ▼♦❞❡r♥❛✳ ✷✵✵✺✳ ✺✼✽♣✳ ❬✶✹❪ ●❖▼❊❙✱ ❆❧❡①❛♥❞r❛❀ ❘❆▲❍❆✱ ❊❧❢r✐❞❛❀ P❆▲❍❆❘❊❙✱ P❡❞r♦✭❖r❣✳✮ ❊❧❡♠❡♥t♦s ❞❡ ▼❛t❡♠át✐❝❛ ♣❛r❛ Pr♦❢❡ss♦r❡s ❞♦ ❊♥s✐♥♦ ❇ás✐❝♦✱ ▲✐s❜♦❛✲P♦rt♦✳ ▲✐❞❡❧✱ ✷✵✵✹✳ ✹✶✸♣✳ ❬✶✺❪ ❘■❇❊■❘❖✱ ❏❛❝❦s♦♥✱ ▼❛t❡♠át✐❝❛✿ ❝✐ê♥❝✐❛✱ ❧✐♥❣✉❛❣❡♠ ❡ t❡❝♥♦❧♦❣✐❛✳ ❙ã♦ P❛✉❧♦✳ ✶✳❡❞✳✱ ❙❝✐♣✐♦♥❡✱ ✷✵✶✵✳ ✸✼✻♣✳ ❬✶✻❪ ❙➱❘●■❖✱ P✳✱ ❋❛t♦s ▼❛t❡♠át✐❝♦s✳ ❉✐s♣♦♥í✈❡❧ ❡♠✿ ❤tt♣✿✴✴❢❛t♦s♠❛t❡♠❛t✐❝♦s✳❜❧♦❣s♣♦t✳❝♦♠✳❜r✴✷✵✶✵✴✵✹✴♦✲✈♦❧✉♠❡✲❞♦✲❞♦❞❡❝❛❡❞r♦✲ r❡❣✉❧❛r✳❤t♠❧✳ ❆❝❡ss♦ ❡♠✿ ✶✻ ❥✉❧✳ ✷✵✶✹✳ ❬✶✼❪ ❙❖❋❚❲❆❘❊ ❈❆❇❘■✱ ❉✐s♣♦♥í✈❡❧ ❡♠✿ ❤tt♣✿✴✴✇✇✇✳❝❛❜r✐✳❝♦♠✴❝❛❜r✐❧♦❣✳❤t♠❧✳ ❆❝❡ss♦ ❡♠✿ ✷✵ ❥✉♥✳ ✷✵✶✹✳ ❬✶✽❪ ❙❖❋❚❲❆❘❊ P❖▲❨✱ ❉✐s♣♦♥í✈❡❧ ❡♠✿ ❤tt♣✿✴✴✇✇✇✳♣❡❞❛✳❝♦♠✴♣♦❧②✴✳ ❆❝❡ss♦ ❡♠✿ ✷✻ ❥✉♥✳ ✷✵✶✹✳ ❬✶✾❪ ❚■❩❩■❖❚❚■✱ ❏♦sé ●✉✐❧❤❡r♠❡✱ ▼❛t❡♠át✐❝❛✳ ❙ã♦ P❛✉❧♦✳ ➪t✐❝❛✱ ✶✾✹✹✳ ✹✾✻✳ ✹✼

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