The Sumerians, one of the earliest civilizations to develop a sophisticated system of mathematics, primarily used a sexagesimal (base 60) numbering system, which was later adopted by the Babylonians. The sexagesimal system is built upon regular numbers, which are constructed by multiplying the prime factors 2, 3, and 5 together in various combinations.
The primary advantage of the sexagesimal system and its reliance on regular numbers is the ease of performing arithmetic operations due to a large number of divisors. The base 60 allowed the Sumerians to represent several fractions more naturally and straightforwardly, making calculations more convenient. The number 60 is a regular number, being divisible by the prime factors 2, 3, and 5. As a result, 60 has many divisors: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. These divisors enable more shared multiples between the numbers, making the system more convenient for performing arithmetic operations.
Their mathematical system was particularly useful in various practical applications such as astronomy, commerce, and civil engineering, where dividing and multiplying values was a common necessity. The use of these regular numbers in their base system made calculations more manageable, especially when conducting tasks like measuring land, calculating weights and quantities, or tracking astronomical observations, which required breaking down units into smaller parts and performing arithmetic operations that involve fractions.
Regular numbers have more shared multiples than non-regular numbers because they have more divisors, which simplifies arithmetic operations. This is particularly useful when dealing with fractions and making calculations in situations where various units need to be broken down and combined. Regular numbers have a very special property: they can be factored into a product of smaller regular numbers. For instance, one of the simplest regular numbers is 6, which can be factored into 2 x 3. Because regular numbers can be factored in this way, they have a large number of integer divisors, which makes multiplication and division by other regular numbers more likely to result in a whole number.
Non-regular numbers, however, are not divisible by any regular numbers other than 1 and themselves. For instance, 7 is only divisible by 1 and 7, and cannot be factored any further. When you multiply or divide non-regular numbers by other non-regular numbers, the resulting values may not have an easily factorizable form. This often results in irrational numbers, which represent numbers that cannot be expressed as a ratio of integers.
Mathematically, multiples of a number are the same as harmonics. So if the harmonics of 1 are 1, 2, 3, 4, 5, 6, 7, 8 etc and the harmonics of 3 and 3, 6, 9, 12, 15, 18 etc then the multiples of those number will be the same as the harmonics.
In music, when two notes in an interval share more harmonics, the human ear perceives the relationship between those notes as more consonant and pleasant. The simple relationships between their frequencies allow our auditory system to process and recognize patterns more easily, leading to a harmonious sensation. Harmonic intervals that contain only regular numbers sound better than ones that do not because they have simple frequency ratios that create shared harmonics, which our auditory system perceives as consonant and pleasing.
In the following image you can see how in the very consonant octave (2:1) every second is shared.
The primary advantage of the sexagesimal system and its reliance on regular numbers is the ease of performing arithmetic operations due to a large number of divisors. The base 60 allowed the Sumerians to represent several fractions more naturally and straightforwardly, making calculations more convenient. The number 60 is a regular number, being divisible by the prime factors 2, 3, and 5. As a result, 60 has many divisors: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. These divisors enable more shared multiples between the numbers, making the system more convenient for performing arithmetic operations.
Their mathematical system was particularly useful in various practical applications such as astronomy, commerce, and civil engineering, where dividing and multiplying values was a common necessity. The use of these regular numbers in their base system made calculations more manageable, especially when conducting tasks like measuring land, calculating weights and quantities, or tracking astronomical observations, which required breaking down units into smaller parts and performing arithmetic operations that involve fractions.
Regular numbers have more shared multiples than non-regular numbers because they have more divisors, which simplifies arithmetic operations. This is particularly useful when dealing with fractions and making calculations in situations where various units need to be broken down and combined. Regular numbers have a very special property: they can be factored into a product of smaller regular numbers. For instance, one of the simplest regular numbers is 6, which can be factored into 2 x 3. Because regular numbers can be factored in this way, they have a large number of integer divisors, which makes multiplication and division by other regular numbers more likely to result in a whole number.
Non-regular numbers, however, are not divisible by any regular numbers other than 1 and themselves. For instance, 7 is only divisible by 1 and 7, and cannot be factored any further. When you multiply or divide non-regular numbers by other non-regular numbers, the resulting values may not have an easily factorizable form. This often results in irrational numbers, which represent numbers that cannot be expressed as a ratio of integers.
Mathematically, multiples of a number are the same as harmonics. So if the harmonics of 1 are 1, 2, 3, 4, 5, 6, 7, 8 etc and the harmonics of 3 and 3, 6, 9, 12, 15, 18 etc then the multiples of those number will be the same as the harmonics.
In music, when two notes in an interval share more harmonics, the human ear perceives the relationship between those notes as more consonant and pleasant. The simple relationships between their frequencies allow our auditory system to process and recognize patterns more easily, leading to a harmonious sensation. Harmonic intervals that contain only regular numbers sound better than ones that do not because they have simple frequency ratios that create shared harmonics, which our auditory system perceives as consonant and pleasing.
In the following image you can see how in the very consonant octave (2:1) every second is shared.
In the next image you can see how in the more dissonant tritone or devils interval (7:5) only every seventh harmonic is shared.
To explain this better, when two or more notes are played simultaneously, their frequencies combine to create a complex wave pattern. When the frequency ratios between the notes are simple, the resulting pattern has a regular and repeating waveform, which our brains find more pleasing and easy to process. For example, the frequency ratio of an octave (2:1) produces a wave pattern where the peaks and troughs of the waves line up every time the frequency doubles, creating a very tight and strong consonance.
In contrast, a frequency ratio of 7:5 creates a wave pattern that is more complex and irregular since the peaks and troughs of the waves do not align as perfectly. This results in a less consonant and less pleasing sound. Therefore, musical intervals that contain only regular numbers (such as the octave, perfect fifth, perfect fourth, etc.) have simple and repetitive waveforms that are more pleasing to the ear than intervals that do not contain regular numbers.
So, in both mathematics and music, the simplicity of the relationships created by shared multiples or harmonics allows for easier arithmetic operations in math and more consonant, pleasant musical intervals. This pattern arises due to the underlying mathematical connection between the two disciplines. So, just as regular numbers create simple, repetitive waveforms in music that are pleasing to the ear, they also create simple and efficient calculations in mathematics that are pleasing to the mind.
In contrast, a frequency ratio of 7:5 creates a wave pattern that is more complex and irregular since the peaks and troughs of the waves do not align as perfectly. This results in a less consonant and less pleasing sound. Therefore, musical intervals that contain only regular numbers (such as the octave, perfect fifth, perfect fourth, etc.) have simple and repetitive waveforms that are more pleasing to the ear than intervals that do not contain regular numbers.
So, in both mathematics and music, the simplicity of the relationships created by shared multiples or harmonics allows for easier arithmetic operations in math and more consonant, pleasant musical intervals. This pattern arises due to the underlying mathematical connection between the two disciplines. So, just as regular numbers create simple, repetitive waveforms in music that are pleasing to the ear, they also create simple and efficient calculations in mathematics that are pleasing to the mind.
Platonic and Archimedean solids are made from only regular numbers. For a regular solid to be formed by assembling identical regular polygons at their corners or vertices, the angles of the polygons must divide exactly into 360 degrees. That means that the sum of the interior angles of each polygon must be a factor of 360 degrees in order for it to be used to construct a regular solid.
For a triangle (3-sided polygon), the sum of the angles is 180 degrees, which is a factor of 360. For a square (4-sided polygon), each interior angle is 90 degrees, which is also a factor of 360. Similarly, the pentagon (5-sided polygon) and hexagon (6-sided polygon) also divide evenly into 360 degrees, making them suitable for use in regular solids.
However, for polygons with 7, 11, or 13 sides, the interior angle measures out to a fraction of a degree when divided into 360. Therefore, it is not possible to use these polygons to construct regular solids, because the angles between the faces cannot be the same around all the vertices of the polyhedron.
Therefore, the angles between the faces of the polygons determine the symmetry and regularity of a solid, and it is the even divisibility of the angles into 360 degrees that allows for regular, symmetrical solids to be formed. Since all divisors of 360, and also 12 and 60 are regular numbers, it makes sense that these solids are constructed using only regular numbers.
For a triangle (3-sided polygon), the sum of the angles is 180 degrees, which is a factor of 360. For a square (4-sided polygon), each interior angle is 90 degrees, which is also a factor of 360. Similarly, the pentagon (5-sided polygon) and hexagon (6-sided polygon) also divide evenly into 360 degrees, making them suitable for use in regular solids.
However, for polygons with 7, 11, or 13 sides, the interior angle measures out to a fraction of a degree when divided into 360. Therefore, it is not possible to use these polygons to construct regular solids, because the angles between the faces cannot be the same around all the vertices of the polyhedron.
Therefore, the angles between the faces of the polygons determine the symmetry and regularity of a solid, and it is the even divisibility of the angles into 360 degrees that allows for regular, symmetrical solids to be formed. Since all divisors of 360, and also 12 and 60 are regular numbers, it makes sense that these solids are constructed using only regular numbers.
Highly totient numbers have more solutions to the equation phi(x) = k than any preceding number (where phi is Euler's totient function). 1, 2, 4, 8, 12, 24, 48, 72, 144, 240, 432, 480, 576, 720, 1152, 1440, 2880, 4320, 5760, 8640, 11520, 17280, 25920, etc.
All highly totient numbers are 5-smooth numbers.
5-smooth numbers: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, 36, 40, 45, 48, 50, 54, 60, 64, 72, 75, 80, 81, 90, 96, 100, 108, 120, 125, 128, 135, 144, 150, 160, 162, 180, 192, 200, 216, 225, 240, 243, 250, 256, 270, 288, 300, 320, 324, 360, 375, 384, 400, 405, 432, etc.
Highly composite numbers have more divisors than any smaller number. 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 7560, 10080, 15120, 20160, 25200, 27720, 45360, 50400, 55440, 83160, 110880, 166320, 221760, 277200, 332640, 498960, 554400, 665280, 720720, 1081080, 1441440, 2162160, etc.
Superior highly composite numbers 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440, 4324320, 21621600, etc.
All highly totient numbers are 5-smooth numbers.
5-smooth numbers: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, 36, 40, 45, 48, 50, 54, 60, 64, 72, 75, 80, 81, 90, 96, 100, 108, 120, 125, 128, 135, 144, 150, 160, 162, 180, 192, 200, 216, 225, 240, 243, 250, 256, 270, 288, 300, 320, 324, 360, 375, 384, 400, 405, 432, etc.
Highly composite numbers have more divisors than any smaller number. 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 7560, 10080, 15120, 20160, 25200, 27720, 45360, 50400, 55440, 83160, 110880, 166320, 221760, 277200, 332640, 498960, 554400, 665280, 720720, 1081080, 1441440, 2162160, etc.
Superior highly composite numbers 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440, 4324320, 21621600, etc.