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Fix multi-digit exponents
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edit approved Oct 22, 2021 at 15:24
Peter Taylor
edit approved Oct 22, 2021 at 15:24
Peter Taylor
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too long for a comment: The exact numbers in @Matt's answers are all algebraic and their minimal polynomials are:

For the area: root of $$81x^9 - 5184x^8 + 60012x^7 + 1080072x^6 - 30787658x^5 + 308408248x^4 - 1618286324x^3 + 4501458360x^2 - 5533131519x + 1109800008$$ near $$0.24546347734508587544309378249...$$

Distance from center: root of $$617673396283947x^36 - 51122005539352848x^32 - 976537068350570496x^28 - 6031070320116105216x^24 + 83285306175293227008x^20 - 71803381414983892992x^16 + 47645844627271974912x^12 - 12502304826882785280x^8 + 600386125323829248x^4 - 4503599627370496$$$$617673396283947x^{36} - 51122005539352848x^{32} - 976537068350570496x^{28} - 6031070320116105216x^{24} + 83285306175293227008x^{20} - 71803381414983892992x^{16} + 47645844627271974912x^{12} - 12502304826882785280x^8 + 600386125323829248x^4 - 4503599627370496$$ near $$3.1583535536268193745169822798...$$ The corresponding value of $b$: root of $$2304x^9 - 2304x^8 - 1536x^7 + 1920x^6 - 1120x^5 - 512x^4 + 128x^3 + 40x^2 + 5x + 1$$ near $$1.2998723034626336283906933307...$$ None of these numbers have a radical expression.

too long for a comment: The exact numbers in @Matt's answers are all algebraic and their minimal polynomials are:

For the area: root of $$81x^9 - 5184x^8 + 60012x^7 + 1080072x^6 - 30787658x^5 + 308408248x^4 - 1618286324x^3 + 4501458360x^2 - 5533131519x + 1109800008$$ near $$0.24546347734508587544309378249...$$

Distance from center: root of $$617673396283947x^36 - 51122005539352848x^32 - 976537068350570496x^28 - 6031070320116105216x^24 + 83285306175293227008x^20 - 71803381414983892992x^16 + 47645844627271974912x^12 - 12502304826882785280x^8 + 600386125323829248x^4 - 4503599627370496$$ near $$3.1583535536268193745169822798...$$ The corresponding value of $b$: root of $$2304x^9 - 2304x^8 - 1536x^7 + 1920x^6 - 1120x^5 - 512x^4 + 128x^3 + 40x^2 + 5x + 1$$ near $$1.2998723034626336283906933307...$$ None of these numbers have a radical expression.

too long for a comment: The exact numbers in @Matt's answers are all algebraic and their minimal polynomials are:

For the area: root of $$81x^9 - 5184x^8 + 60012x^7 + 1080072x^6 - 30787658x^5 + 308408248x^4 - 1618286324x^3 + 4501458360x^2 - 5533131519x + 1109800008$$ near $$0.24546347734508587544309378249...$$

Distance from center: root of $$617673396283947x^{36} - 51122005539352848x^{32} - 976537068350570496x^{28} - 6031070320116105216x^{24} + 83285306175293227008x^{20} - 71803381414983892992x^{16} + 47645844627271974912x^{12} - 12502304826882785280x^8 + 600386125323829248x^4 - 4503599627370496$$ near $$3.1583535536268193745169822798...$$ The corresponding value of $b$: root of $$2304x^9 - 2304x^8 - 1536x^7 + 1920x^6 - 1120x^5 - 512x^4 + 128x^3 + 40x^2 + 5x + 1$$ near $$1.2998723034626336283906933307...$$ None of these numbers have a radical expression.

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Moritz Firsching
Moritz Firsching
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too long for a comment: The exact numbers in @Matt's answers are all algebraic and their minimal polynomials are:

For the area: root of $$81x^9 - 5184x^8 + 60012x^7 + 1080072x^6 - 30787658x^5 + 308408248x^4 - 1618286324x^3 + 4501458360x^2 - 5533131519x + 1109800008$$ near $$0.24546347734508587544309378249...$$

Distance from center: root of $$617673396283947x^36 - 51122005539352848x^32 - 976537068350570496x^28 - 6031070320116105216x^24 + 83285306175293227008x^20 - 71803381414983892992x^16 + 47645844627271974912x^12 - 12502304826882785280x^8 + 600386125323829248x^4 - 4503599627370496$$ near $$3.1583535536268193745169822798...$$ The corresponding value of $b$: root of $$2304x^9 - 2304x^8 - 1536x^7 + 1920x^6 - 1120x^5 - 512x^4 + 128x^3 + 40x^2 + 5x + 1$$ near $$1.2998723034626336283906933307...$$ None of these numbers have a radical expression.