My sorted posts, answers, comments in MEDIUM, the platform, on perfect numbers, and on tetrahedrons, icosahedrons, and cubes that indwell perfect numbers

Perfect numbers are exactly filled up, and filled in, by equal sized tetrahedrons, which are pyramids. Perfect numbers are also filled in by cubes. 

Take 496 for example, a perfect number.

If you have 496 tetrahedrons, they are altogether made of 1992 triangles (1600+360+32). 

496 tetrahedrons also make 24, 20-component icosahedrons, with 16 additional tetrahedrons.

In cubes, 496 is the sum of the first several odd-edged cubes.

One(1) cube plus(+) 27 cubes(3×9 cubes) plus(+) 125 cubes(5×25 cubes) plus(+)343 cubes(7×49 cubes).

1+27+125+343=496. These internal perfect cubes(1,27,125,343), viewed as components of perfect numbers, added together and sumed, are the perfect number 496. And not just 496, but this is true of all perfect numbers, that, internally, they are sets of perfect cube-clusteres, clustered together into perfect cubes, if you can see them that way. 

Whether tetrahedrons or cubes, they pack tightly. 

Each tetrahedron is made of four equilateral triangles🔺. This exact filling of perfect numbers with pyramidal tetrahedrons scales to infinity.

 A tetrahedron is [four] equilateral triangles🔺🔺🔺🔺 (of equal lines) folded concentrically together into a pyramidal tetrahedron. 

The fact that perfect numbers divide by 4, again and again, until they come to four(4) equal Mersenne primes is no coincidence. Example: 496 ÷4 = 124. 124÷4= 31.

31 is a Mersenne prime.

24 icosahedrons, which are spherical clusters of 20 tetrahedrons, represent 480 tetrahedrons of 496 tetrahedrons. 16 more pyramidal tetrahedrons make the perfect number 496 with no dangling fraction or fragment.

 Structurally, there can be no phantom odd perfect unless, you can make a 4 a 3, or a 3 into a 4, or a tetrahedron with 3 triangles instead of 4.

People speculate that there might be some astronomically big perfect number that could be odd because they see no geometrical structure filling them.

 They seem mysterious and become astronomical in size as the base of a tetrahedron is larger than the apex tip.

Perfect numbers stack, tessellate, and ring as triangles and inverted triangles. 

Perfect numbers as tetrahedrons cluster into icosahedrons.

This is hard to show here in text. 

Try to picture upright and inverted triangles stacked or tessellated tightly as a grid of triangles.

__________∆

_______.∆∆∆∆

_____∆∆∆∆∆∆∆

Some see perfect numbers as triangular by looking at stacks of balls, but they are not as tightly packed as tetrahedrons clustered into icosahedrons.

There is internal structure in all perfect numbers forcing them to be even.

 Internally, perfect numbers are all either an odd multiple of 7 plus 1, or and even multiple of 7 plus 6. Either way, this forces an even number.

Examples: 

A) The "perfect" 496 is 490 +6.

490 is a even multiple of 7 (7×70).

B) The "perfect" 8128 is 8127+1.

 8127 is an odd multiple of 7 (7000+1120+7).

C) The "perfect" 33,550,336 is 33,550,335 +1.

33,550,335 is a multiple of 7 (28million+4.9m+630k+14k+6,300+35).

Perfect numbers ALL divide by four (4), and geometrically form an exactitude of tetrahedrons, which are four triangles each, except the starter 6, which can be viewed as either 

6 triangles🔻🔺🔻🔺🔻🔺, 

1 tetrahedron ∆ and two triangles🔻🔺,

 (I could not find a tetrahedron emoji.)

 or as 6 tetrahedrons ∆∆∆∆∆∆, 

or as

 6 squares⬜⬜⬜⬜⬜⬜,

 1 cube📦, or 

 6 cubes📦📦📦📦📦📦. 

Four(4) sided tetrahedrons exactly fill in all perfect numbers.

 With any perfect number, you can substitute tetrahedrons for each integer.

For example: 496 tetrahedrons.

FibetyJibets, August 11, 2026

😐CUBE NOTES ADDED.

Perfect numbers are exactly filled up, and filled in, by equal sized tetrahedrons, which are pyramids. Perfect numbers are also filled in by cubes. 

Take 496 for example, a perfect number.

If you have 496 tetrahedrons, they are altogether made of 1992 triangles (1600+360+32). 

496 tetrahedrons also make 24, 20-component icosahedrons, with 16 additional tetrahedrons.

In cubes, 496 is the sum of the first several odd-edged cubes(1,3,5,7.), which are the 3×3×3 edges of the 27 cubes CUBE, which has 1 center cube in a 26 cubes shell, the 5×5×5 edges of the 125 cubes CUBE, which has a center cube of 27 cubes, in a shell of 98  cubes, and the 7×7×7 edges of the 343 cubes CUBE, which is a 125 cubes CUBE in a 218 cubes shell. 

One(1) cube plus(+) 27 cubes(3×9 cubes) plus(+) 125 cubes(5×25 cubes) plus(+)343 cubes(7×49 cubes).

1+27+125+343=496. These internal perfect cubes(1,27,125,343), viewed as components of perfect numbers, added together and sumed, are the perfect number 496. And not just 496, but this is true of all perfect numbers, that, internally, they are sets of perfect cube-clusteres, clustered together into perfect cubes, if you can see them that way. 

Another example is the perfect number 8128. 

8128 is 127(a Mersenne prime) times 64.

64 cubes is 4×4×4, which makes a perfect cube. 127 of these perfect cube sets makes 8128 64-cube, CUBE-sets.  125 is a perfect cube, therefore, since 125×64=8000, which is a perfect cube, and 2 more 64 cube CUBES is another 128 cubes, and since 8000+128=8128(a perfect number), it is therefore a fact that 127×64 makes a perfect number of perfect cubes , with one 8000 cube, perfect cube (125, 64-cube CUBES), with two extra 64-cube CUBES .

Whether tetrahedrons or cubes, they pack tightly. 

Each tetrahedron is made of four equilateral triangles🔺. This exact filling of perfect numbers with pyramidal tetrahedrons scales to infinity.

 A tetrahedron is [four] equilateral triangles🔺🔺🔺🔺 (of equal lines) folded concentrically together into a pyramidal tetrahedron. 

The fact that perfect numbers divide by 4, again and again, until they come to four(4) equal Mersenne primes is no coincidence. Example: 496 ÷4 = 124. 124÷4= 31.

31 is a Mersenne prime.

24 icosahedrons, which are spherical clusters of 20 tetrahedrons, represent 480 tetrahedrons of 496 tetrahedrons. 16 more pyramidal tetrahedrons make the perfect number 496 with no dangling fraction or fragment.

 Structurally, there can be no phantom odd perfect unless, you can make a 4 a 3, or a 3 into a 4, or a tetrahedron with 3 triangles instead of 4.

People speculate that there might be some astronomically big perfect number that could be odd because they see no geometrical structure filling them.

 They seem mysterious and become astronomical in size as the base of a tetrahedron is larger than the apex tip.

Perfect numbers stack, tessellate, and ring as triangles and inverted triangles. 

Perfect numbers as tetrahedrons cluster into icosahedrons.

This is hard to show here in text. 

Try to picture upright and inverted triangles stacked or tessellated tightly as a grid of triangles.

__________∆

_______.∆∆∆∆

_____∆∆∆∆∆∆∆

Some see perfect numbers as triangular by looking at stacks of balls, but they are not as tightly packed as tetrahedrons clustered into icosahedrons.

There is internal structure in all perfect numbers forcing them to be even.

 Internally, perfect numbers are all either an odd multiple of 7 plus 1, or and even multiple of 7 plus 6. Either way, this forces an even number.

Examples: 

A) The "perfect" 496 is 490 +6.

490 is a even multiple of 7 (7×70).

B) The "perfect" 8128 is 8127+1.

 8127 is an odd multiple of 7 (7000+1120+7).

C) The "perfect" 33,550,336 is 33,550,335 +1.

33,550,335 is a multiple of 7 (28million+4.9m+630k+14k+6,300+35).

Perfect numbers ALL divide by four (4), and geometrically form an exactitude of tetrahedrons, which are four triangles each, except the starter 6, which can be viewed as either 

6 triangles🔻🔺🔻🔺🔻🔺, 

1 tetrahedron ∆ and two triangles🔻🔺,

 (I could not find a tetrahedron emoji.)

 or as 6 tetrahedrons ∆∆∆∆∆∆, 

or as

 6 squares⬜⬜⬜⬜⬜⬜,

 1 cube📦, or 

 6 cubes📦📦📦📦📦📦. 

Four(4) sided tetrahedrons exactly fill in all perfect numbers.

 With any perfect number, you can substitute tetrahedrons for each integer.

For example: 496 tetrahedrons.

FibetyJibets, August 11, 2026