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ƧƿѦςɛ♏ѦਹѤʞ @[email protected] · 2024-12-24 · 16:58 UTC

Nicomachus's theorem:
https://www.youtube.com/watch?v=WY5X_3q80WY
#maths #mathematics #squares #cubes

#maths #mathematics #squares #cubes

Alan J. Cain @[email protected] · 2026-02-05 · 11:09 UTC

In later antiquity and the middle ages, a ‘circular number’ was one that reappeared in its own powers: 5 and 6 were circular number since their powers (25, 125, 625, ...; 36, 216, 1296, ...) always end in 5 or 6.

Nicomachus (fl. c.100 CE), Proclus (410/12–485 CE), and Boethius (c.480–c.524 CE) discussed them. In an educational textbook, Cassiodorus (c.485–c.585 CE) gave this definition:

‘A circular number is one that when it is multiplied by itself, beginning from itself turns back to itself, for example 5 times 5 is 25 *as the diagram indicates*’. (emphasis added; see 1st+2nd attached images)

So circular numbers seem to have been a connection between number symbolism and a geometrical aesthetic admiration of circles and spheres (more on this in a later post).

5 being a circular number crops up in the in the late mediaeval poem ‘Sir Gawain and the Green Knight’ (c. late 13th century). 5 was used as symbol of perfection and eternity: Gawain's virtues were five and many times five, and they were linked to the pentagram, the five-pointed star, which was his emblem. At line $625 = 5 \times 5 \times 5 \times 5$, the poet says that the pentagram was a symbol set up by Solomon; it was known as ‘þe endeles knot’. This name presumably refers to how the pentagram can be drawn in a single unbroken stroke (see 3rd attached image)

Very subtly, the circularity is hinted at by the first line of the poem (‘Siþen þe sege and þe assaut watz sesed at Troye’) being echoed at line 2525 — or 25-25 — (‘After þe segge and þe asaute watz sesed at Troye’).

1/2

#NumberSymbolism #arithmology #Nicomachus #Proclus #Boethius #Cassiodorus #poetry

#numbersymbolism #arithmology #nicomachus #proclus #boethius #cassiodorus

Alan J. Cain @[email protected] · 2026-02-05 · 11:09 UTC

In later antiquity and the middle ages, a ‘circular number’ was one that reappeared in its own powers: 5 and 6 were circular number since their powers (25, 125, 625, ...; 36, 216, 1296, ...) always end in 5 or 6.

Nicomachus (fl. c.100 CE), Proclus (410/12–485 CE), and Boethius (c.480–c.524 CE) discussed them. In an educational textbook, Cassiodorus (c.485–c.585 CE) gave this definition:

‘A circular number is one that when it is multiplied by itself, beginning from itself turns back to itself, for example 5 times 5 is 25 *as the diagram indicates*’. (emphasis added; see 1st+2nd attached images)

So circular numbers seem to have been a connection between number symbolism and a geometrical aesthetic admiration of circles and spheres (more on this in a later post).

5 being a circular number crops up in the in the late mediaeval poem ‘Sir Gawain and the Green Knight’ (c. late 13th century). 5 was used as symbol of perfection and eternity: Gawain's virtues were five and many times five, and they were linked to the pentagram, the five-pointed star, which was his emblem. At line $625 = 5 \times 5 \times 5 \times 5$, the poet says that the pentagram was a symbol set up by Solomon; it was known as ‘þe endeles knot’. This name presumably refers to how the pentagram can be drawn in a single unbroken stroke (see 3rd attached image)

Very subtly, the circularity is hinted at by the first line of the poem (‘Siþen þe sege and þe assaut watz sesed at Troye’) being echoed at line 2525 — or 25-25 — (‘After þe segge and þe asaute watz sesed at Troye’).

1/2

#NumberSymbolism #arithmology #Nicomachus #Proclus #Boethius #Cassiodorus #poetry

#poetry #cassiodorus #boethius #proclus #nicomachus #arithmology

Alan J. Cain @[email protected] · 2026-02-05 · 11:09 UTC

In later antiquity and the middle ages, a ‘circular number’ was one that reappeared in its own powers: 5 and 6 were circular number since their powers (25, 125, 625, ...; 36, 216, 1296, ...) always end in 5 or 6.

Nicomachus (fl. c.100 CE), Proclus (410/12–485 CE), and Boethius (c.480–c.524 CE) discussed them. In an educational textbook, Cassiodorus (c.485–c.585 CE) gave this definition:

‘A circular number is one that when it is multiplied by itself, beginning from itself turns back to itself, for example 5 times 5 is 25 *as the diagram indicates*’. (emphasis added; see 1st+2nd attached images)

So circular numbers seem to have been a connection between number symbolism and a geometrical aesthetic admiration of circles and spheres (more on this in a later post).

5 being a circular number crops up in the in the late mediaeval poem ‘Sir Gawain and the Green Knight’ (c. late 13th century). 5 was used as symbol of perfection and eternity: Gawain's virtues were five and many times five, and they were linked to the pentagram, the five-pointed star, which was his emblem. At line $625 = 5 \times 5 \times 5 \times 5$, the poet says that the pentagram was a symbol set up by Solomon; it was known as ‘þe endeles knot’. This name presumably refers to how the pentagram can be drawn in a single unbroken stroke (see 3rd attached image)

Very subtly, the circularity is hinted at by the first line of the poem (‘Siþen þe sege and þe assaut watz sesed at Troye’) being echoed at line 2525 — or 25-25 — (‘After þe segge and þe asaute watz sesed at Troye’).

1/2

#NumberSymbolism #arithmology #Nicomachus #Proclus #Boethius #Cassiodorus #poetry

#poetry #cassiodorus #boethius #proclus #nicomachus #arithmology

Alan J. Cain @[email protected] · 2026-02-05 · 11:09 UTC

In later antiquity and the middle ages, a ‘circular number’ was one that reappeared in its own powers: 5 and 6 were circular number since their powers (25, 125, 625, ...; 36, 216, 1296, ...) always end in 5 or 6.

Nicomachus (fl. c.100 CE), Proclus (410/12–485 CE), and Boethius (c.480–c.524 CE) discussed them. In an educational textbook, Cassiodorus (c.485–c.585 CE) gave this definition:

‘A circular number is one that when it is multiplied by itself, beginning from itself turns back to itself, for example 5 times 5 is 25 *as the diagram indicates*’. (emphasis added; see 1st+2nd attached images)

So circular numbers seem to have been a connection between number symbolism and a geometrical aesthetic admiration of circles and spheres (more on this in a later post).

5 being a circular number crops up in the in the late mediaeval poem ‘Sir Gawain and the Green Knight’ (c. late 13th century). 5 was used as symbol of perfection and eternity: Gawain's virtues were five and many times five, and they were linked to the pentagram, the five-pointed star, which was his emblem. At line $625 = 5 \times 5 \times 5 \times 5$, the poet says that the pentagram was a symbol set up by Solomon; it was known as ‘þe endeles knot’. This name presumably refers to how the pentagram can be drawn in a single unbroken stroke (see 3rd attached image)

Very subtly, the circularity is hinted at by the first line of the poem (‘Siþen þe sege and þe assaut watz sesed at Troye’) being echoed at line 2525 — or 25-25 — (‘After þe segge and þe asaute watz sesed at Troye’).

1/2

#NumberSymbolism #arithmology #Nicomachus #Proclus #Boethius #Cassiodorus #poetry

#poetry #cassiodorus #boethius #proclus #nicomachus #arithmology

Alan J. Cain @[email protected] · 2026-02-05 · 11:09 UTC

In later antiquity and the middle ages, a ‘circular number’ was one that reappeared in its own powers: 5 and 6 were circular number since their powers (25, 125, 625, ...; 36, 216, 1296, ...) always end in 5 or 6.

Nicomachus (fl. c.100 CE), Proclus (410/12–485 CE), and Boethius (c.480–c.524 CE) discussed them. In an educational textbook, Cassiodorus (c.485–c.585 CE) gave this definition:

‘A circular number is one that when it is multiplied by itself, beginning from itself turns back to itself, for example 5 times 5 is 25 *as the diagram indicates*’. (emphasis added; see 1st+2nd attached images)

So circular numbers seem to have been a connection between number symbolism and a geometrical aesthetic admiration of circles and spheres (more on this in a later post).

5 being a circular number crops up in the in the late mediaeval poem ‘Sir Gawain and the Green Knight’ (c. late 13th century). 5 was used as symbol of perfection and eternity: Gawain's virtues were five and many times five, and they were linked to the pentagram, the five-pointed star, which was his emblem. At line $625 = 5 \times 5 \times 5 \times 5$, the poet says that the pentagram was a symbol set up by Solomon; it was known as ‘þe endeles knot’. This name presumably refers to how the pentagram can be drawn in a single unbroken stroke (see 3rd attached image)

Very subtly, the circularity is hinted at by the first line of the poem (‘Siþen þe sege and þe assaut watz sesed at Troye’) being echoed at line 2525 — or 25-25 — (‘After þe segge and þe asaute watz sesed at Troye’).

1/2

#NumberSymbolism #arithmology #Nicomachus #Proclus #Boethius #Cassiodorus #poetry

#poetry #cassiodorus #boethius #proclus #nicomachus #arithmology

Alan J. Cain @[email protected] · 2026-02-01 · 21:22 UTC

For each day of February, I intend to post a short interesting story/image/fact/anecdote related to the aesthetics of mathematics.

February has 28 days, and 28 is the second perfect number, so let's start there.

Definitions first: a ‘perfect’ number is equal to the sum of its own proper divisors (28 = 1+2+4+7+14). If the sum of proper divisors is greater than the number itself, the number is ‘abundant’; if less, it is ‘deficient’.

Philo of Alexandria (fl. early 1st century CE) seems to have connected perfect numbers with beauty, at least via the order emplaced in the world during the Biblical 6 (= 1+2+3) days of creation. (The perfect number of days of the hexaëmeron was emphasized by later writers like Methodius of Olympus (d. c.310 CE) and Augustine of Hippo (354–430 CE), though not in explicitly aesthetic terms.)

Nicomachus of Gerasa (fl. 100 CE) seems to have thought that the (rare) perfect numbers were beautiful, and that the much more common abundant and deficient numbers were ugly, likening them to monstrous creatures with too many or too few limbs, mouths, eyes.

Boethius (c.480–c.524 CE) agreed with Nicomachus and was even more specific about the parallel to monsters: deficient numbers were like the one-eyed Cyclopes; abundant numbers were like the triple-headed or -bodied Geryon (image attached).

1/2

#MathematicalBeauty #MathHist #PerfectNumber #aesthetics

#aesthetics #perfectnumber #mathhist #mathematicalbeauty

Alan J. Cain @[email protected] · 2026-02-01 · 21:22 UTC

For each day of February, I intend to post a short interesting story/image/fact/anecdote related to the aesthetics of mathematics.

February has 28 days, and 28 is the second perfect number, so let's start there.

Definitions first: a ‘perfect’ number is equal to the sum of its own proper divisors (28 = 1+2+4+7+14). If the sum of proper divisors is greater than the number itself, the number is ‘abundant’; if less, it is ‘deficient’.

Philo of Alexandria (fl. early 1st century CE) seems to have connected perfect numbers with beauty, at least via the order emplaced in the world during the Biblical 6 (= 1+2+3) days of creation. (The perfect number of days of the hexaëmeron was emphasized by later writers like Methodius of Olympus (d. c.310 CE) and Augustine of Hippo (354–430 CE), though not in explicitly aesthetic terms.)

Nicomachus of Gerasa (fl. 100 CE) seems to have thought that the (rare) perfect numbers were beautiful, and that the much more common abundant and deficient numbers were ugly, likening them to monstrous creatures with too many or too few limbs, mouths, eyes.

Boethius (c.480–c.524 CE) agreed with Nicomachus and was even more specific about the parallel to monsters: deficient numbers were like the one-eyed Cyclopes; abundant numbers were like the triple-headed or -bodied Geryon (image attached).

1/2

#MathematicalBeauty #MathHist #PerfectNumber #aesthetics

#aesthetics #perfectnumber #mathhist #mathematicalbeauty

Alan J. Cain @[email protected] · 2026-02-01 · 21:22 UTC

For each day of February, I intend to post a short interesting story/image/fact/anecdote related to the aesthetics of mathematics.

February has 28 days, and 28 is the second perfect number, so let's start there.

Definitions first: a ‘perfect’ number is equal to the sum of its own proper divisors (28 = 1+2+4+7+14). If the sum of proper divisors is greater than the number itself, the number is ‘abundant’; if less, it is ‘deficient’.

Philo of Alexandria (fl. early 1st century CE) seems to have connected perfect numbers with beauty, at least via the order emplaced in the world during the Biblical 6 (= 1+2+3) days of creation. (The perfect number of days of the hexaëmeron was emphasized by later writers like Methodius of Olympus (d. c.310 CE) and Augustine of Hippo (354–430 CE), though not in explicitly aesthetic terms.)

Nicomachus of Gerasa (fl. 100 CE) seems to have thought that the (rare) perfect numbers were beautiful, and that the much more common abundant and deficient numbers were ugly, likening them to monstrous creatures with too many or too few limbs, mouths, eyes.

Boethius (c.480–c.524 CE) agreed with Nicomachus and was even more specific about the parallel to monsters: deficient numbers were like the one-eyed Cyclopes; abundant numbers were like the triple-headed or -bodied Geryon (image attached).

1/2

#MathematicalBeauty #MathHist #PerfectNumber #aesthetics

#aesthetics #perfectnumber #mathhist #mathematicalbeauty

Alan J. Cain @[email protected] · 2026-02-01 · 21:22 UTC

For each day of February, I intend to post a short interesting story/image/fact/anecdote related to the aesthetics of mathematics.

February has 28 days, and 28 is the second perfect number, so let's start there.

Definitions first: a ‘perfect’ number is equal to the sum of its own proper divisors (28 = 1+2+4+7+14). If the sum of proper divisors is greater than the number itself, the number is ‘abundant’; if less, it is ‘deficient’.

Philo of Alexandria (fl. early 1st century CE) seems to have connected perfect numbers with beauty, at least via the order emplaced in the world during the Biblical 6 (= 1+2+3) days of creation. (The perfect number of days of the hexaëmeron was emphasized by later writers like Methodius of Olympus (d. c.310 CE) and Augustine of Hippo (354–430 CE), though not in explicitly aesthetic terms.)

Nicomachus of Gerasa (fl. 100 CE) seems to have thought that the (rare) perfect numbers were beautiful, and that the much more common abundant and deficient numbers were ugly, likening them to monstrous creatures with too many or too few limbs, mouths, eyes.

Boethius (c.480–c.524 CE) agreed with Nicomachus and was even more specific about the parallel to monsters: deficient numbers were like the one-eyed Cyclopes; abundant numbers were like the triple-headed or -bodied Geryon (image attached).

1/2

#MathematicalBeauty #MathHist #PerfectNumber #aesthetics

#mathematicalbeauty #mathhist #perfectnumber #aesthetics

Alan J. Cain @[email protected] · 2026-02-01 · 21:22 UTC

For each day of February, I intend to post a short interesting story/image/fact/anecdote related to the aesthetics of mathematics.

February has 28 days, and 28 is the second perfect number, so let's start there.

Definitions first: a ‘perfect’ number is equal to the sum of its own proper divisors (28 = 1+2+4+7+14). If the sum of proper divisors is greater than the number itself, the number is ‘abundant’; if less, it is ‘deficient’.

Philo of Alexandria (fl. early 1st century CE) seems to have connected perfect numbers with beauty, at least via the order emplaced in the world during the Biblical 6 (= 1+2+3) days of creation. (The perfect number of days of the hexaëmeron was emphasized by later writers like Methodius of Olympus (d. c.310 CE) and Augustine of Hippo (354–430 CE), though not in explicitly aesthetic terms.)

Nicomachus of Gerasa (fl. 100 CE) seems to have thought that the (rare) perfect numbers were beautiful, and that the much more common abundant and deficient numbers were ugly, likening them to monstrous creatures with too many or too few limbs, mouths, eyes.

Boethius (c.480–c.524 CE) agreed with Nicomachus and was even more specific about the parallel to monsters: deficient numbers were like the one-eyed Cyclopes; abundant numbers were like the triple-headed or -bodied Geryon (image attached).

1/2

#MathematicalBeauty #MathHist #PerfectNumber #aesthetics

#aesthetics #perfectnumber #mathhist #mathematicalbeauty