There is a well-known formula for the Fibonacci numbers

where

is the golden ratio.

It turns out that there is a way to find for when is not an integer, but the values are complex rather than real.

We have

and we could consider the general `primary’ root of 1 to be defined using

where

and

and

is the circle constant (so ).

Then we have

and so

Thus we arrive at the formula for :

i.e.

Similarly, we can obtain a formula for a generalisation of the Lucas numbers :

generalising

The following are well-known:

A little algebra can be used to show that:

and

so

Similarly,

If, instead, we generalise to the real-valued function :

Then we can show that:

and

so

i.e.

Similarly, we could define

and similar results follow.

]]>I will also show the formulae in action with a worked example.

In the main article, I show that

This formula is essentially a polynomial of rising factorial powers.

Perhaps the most important and useful formulae from the main article are

and

Less important, though equally simple, is that these formulae form a sequence that continues, essentially:

and so on…

The notation here is outlined below.

(There are also alternative formulae in terms of Stirling set numbers. However, the above formulae in terms of Eulerian numbers are perhaps simpler, and utilise simpler coefficients.)

The sequence of cubes begins

1, 8, 27, 64, 125, 216, 343, 512, 729, 1000,…

The sequence of sums of cubes therefore begins

1, 9, 36, 100, 225, 441, 784, 1296, 2025, 3025, …

And we can go further, and sum those!:

1, 10, 46, 146, 371, 812, 1596, 2892, 4917, 7942,…

What is the 1000th term in each of these sequences?

Calculating these by brute force, we obtain, respectively

1000 000 000,

250 500 250 000,

50 250 416 916 700

What is the 1000 000th term in each of these sequences?

Good luck with brute force.

So, we turn to the formulae:

That is,

But then we also have:

Plugging n=1000 into these formulae we obtain:

1000 000 000,

250 500 250 000,

50 250 416 916 700

Plugging n=1000 000 into these formulae we obtain:

1000 000 000 000 000 000,

250 000 500 000 250 000 000 000,

50 000 250 000 416 666 916 666 700 000

The indirect formulae above may be used to calculate the sums of powers simply. However, if a conventional polynomial (of `ordinary’ rather than rising factorial) powers is needed, then the `indirect’ formulae may be expanded to the following `direct’ formulae:

where

where

and

where

(There are also alternative formulae in terms of Stirling set numbers.)

Hypertetrahedral numbers, or permutations-with-replacement:

The binomial coefficients:

The Eulerian numbers:

The unsigned (or signless) Stirling numbers of the first kind, or Stirling cycle numbers are denoted by:

(Please see the main article for the closed formulae defining these.)

The notation

is a -iterated summation.

(See the main article for the details, and formal definition.)

]]>

The first few sums of powers are

where is my notation for the -th triangular number.

Faulhaber’s formula for the general sum of powers is

where the are the Bernoulli numbers (which are rational numbers)

with the convention that .

The first few values include:

,

,

,

for all ,

,

,

,

,

, and

.

(Beware that, in the literature, there are various alternative definitions of the Bernoulli numbers, including the which are the same except that . Other definitions have their as our . However, in this document, I shall only need the , as the most convenient for my purpose.)

The powers may be constructed from hypertetrahedral numbers, which are defined by:

% , , or :

The is my (non-standard) notation for the -th hypertetrahedral number of dimension .

Note the special cases of , the ordinary triangular numbers; ; (for ).

For example, square numbers (i.e. powers of order 2) may be thought of as a sum of two non-overlapping triangular numbers; cubes (i.e. powers of order 3) as a sum of six tetrahedral triangular numbers.

The coefficients of these `triangular polynomials’ are the Eulerian numbers (see below).

Alternatively, square numbers may be thought of as a sum of two overlapping triangular numbers; cubes as a sum of six overlapping tetrahedral triangular numbers.

The coefficients of these are closely related to the Stirling set numbers (see below).

Now the wonderful thing about the hypertetrahedral numbers is that they are extraordinarily easy to sum:

This may even be considered to be an alternative way of defining the hypertetrahedral numbers.

My approach will even allow iterated summations to be carried out directly.

In this post, I will use the convention

so that for all .

The binomial coefficients, which are (positive) integers, are:

These binomial coefficients may be viewed in a triangle (for ), called Pascal’s triangle. The rows of this triangle are:

1;

1, 1;

1, 2, 1;

1, 3, 3, 1;

1, 4, 6, 4, 1;

1, 5, 10, 10, 5, 1;…

Related to Stirling numbers of the second kind are the numbers

% :

[In this case, the notation is mine, and is not standard.]

The Stirling numbers of the second kind, or Stirling set numbers, which are (non-negative) integers, are:

i.e.

The Stirling set numbers may be viewed in a triangle similar to Pascal’s.

1;

0, 1;

0, 1, 1;

0, 1, 3, 1;

0, 1, 7, 6, 1;

0, 1, 15, 25, 10, 1; …

The unsigned (or signless) Stirling numbers of the first kind, or Stirling cycle numbers, which are (obviously non-negative) integers, are:

Again, the Stirling cycle numbers may be viewed in a triangle.

1;

0, 1;

0, 1; 1;

0, 2, 3, 1;

0, 6, 11, 6, 1;

0, 24, 50, 35, 10, 1; …

The Eulerian numbers, which again are (positive) integers, are:

And again, the Eulerian numbers may be viewed in a triangle.

1;

1, 1;

1, 4, 1;

1, 11, 11, 1;

1, 26, 66, 26, 1;

1, 57, 302, 302, 57, 1; …

Here, I would prefer to use a double-struck (or blackboard bold) variant of a majuscule Greek letter sigma (), but am not able to typeset that symbol here, so instead I use a large circled plus ().

I introduce the notation:

for a -iterated summation.

For example

is not summing at all (no iterations);

has one iteration, that is, it is an `ordinary’ sum;

has two iterations, and is a `double’ sum.

has three iterations, and is a `triple’ sum. And so on…

Generally, we define:

Considering powers as sums of overlapping hypertetrahedral numbers of the same order, but different dimensions, we have, for

The lower bound can be adjusted so that the expression is also valid for the special case . Thus, for we have the more convenient

On the other hand, considering powers as sums of non-overlapping hypertetrahedral numbers of different orders, but the same dimension, we also have, for

Here, the upper bound can be adjusted so that the expression is also valid for the special case . Thus, for we have

Recall that

This is the key to the manipulation of the new expressions.

Now recall that

To take sums, we simply adjust the dimension of the hypertetrahedral term thus:

Notice that the only change in the right-hand side is that the factor has become .

And we can even take sums of these, similarly and simply:

where, in the right-hand side, that factor has now become .

And so on…

Likewise, recall also that

To take sums, again, we simply adjust the dimension of the hypertetrahedral term thus:

Similarly, notice that the only change in the right-hand side is that the factor has become .

And again we can take sums of these:

where, in the right-hand side, that factor has now become .

Again, and so on…

The real beauty of these formulae is the similarity between the summation levels. The formulae for and those for (and even etc.) are clearly and simply related.

Generally, we have the formulae (for ):

We could stop there, if we wished. That is, these formulae might already be sufficiently convenient for the calculation of sums of powers.

These forms are essentially polynomials of rising factorial powers, as opposed to polynomials of `ordinary’ powers.

For a fairer comparison with the Faulhaber formula, we can expand the new expressions to form explicit `ordinary’ polymonials.

Following some expansion, and exchanges of order of sums, we find that:

where

and also

where

Of course, for all , and , thus we do not really require separate notations for these. I have done so in an attempt to maintain clarity in the following.

In particular, for a single (one-iteration) sum ():

where

and

where

But, this method also gives us the double (two-iteration) sum ():

where

and

where

and similarly for higher-iteration sums.

When we consider the special case , we find that these expressions are equal:

where the last two are the Kroneker delta function, and an Iverson bracket.

With a substitution in the indexing we can re-write Faulhaber’s formula as

i.e., we have

where

So, when we consider the special case , we find that these expressions are equal:

As an aside, we also have

for any sufficiently large , i.e. .

]]>where

However, I was surprised to find that there are also product formulae involving trigonometric functions.

There is a formula for the number of ways that dominoes can be placed to cover an draughts board:

(Here, would be given by .)

As it happens that

then with a little simple manipulation, this gives, for , various product formulae for the Fibonacci numbers

These are all equivalent for two reasons. The first two have the same terms because for integers , so the upper bounds are the same. Similarly, the last two have the same terms because for integers – 1, so the upper bounds are again the same. Finally, when the first two forms contain an additional term, that term equals 1, because , and this has no effect on the product.

(These formulae do not work for , where the value 1 is given, for the empty product, whereas the true value of is 0.)

Examples using the last form:

]]>My requirements included:

- The formulae should all be of a similar form.
- Where there is a change of dimension, formulae should be given both in terms of the source and the target dimensons.
- No formula should have a surd (root) in the denominator.
- The terms in a surd should have reduced factors. (So, in particular, any integer under a square root should be square-free.)

I use the symbols:

a = length of an edge

r = inradius, i.e. the radius of the insphere

ρ (rho) = midradius, i.e. the radius of the midsphere

R = circumradius, i.e. the radius of the circumsphere

S = total surface area of polyhedron

C = volume or capacity of polyhedron

The tables were generated with the help of *Maxima* and *Python*. Much of the work to produce the final forms required hand calculation. The final tables have been checked by computer for correctness.

It is possible that these formulae are not the simplest possible, if I have missed an available simplification.

For example, it happens that:

(a + b√c)² = h + k·√c

where

h = a² + b²c

k = 2ab

so

h + k·√c = (a + b√c)²

where, solving for a and b,

a = ±√( (h ± √(h²−k²c)) / 2 )

b = ±√( (h ± √(h²−k²c)) / 2c )

and so it is possible that a form such as

√( h + k·√c )

might simplify to

a + b√c

for certain integers c, h and k.

My formulae include consideration for the following spheres:

- The insphere is the largest that fits inside the polyhedron; it touches the centres of the faces.
- The midsphere passes through the midpoints of the edges.
- The circumsphere is the smallest that fits around the polyhedron; it passes through the vertices.

The formulae are for the radius of each of these spheres. I do not calculate the volumes or surface areas of these spheres, as those are related directly to the radii by the formulae

Area = 2τ rad²

Vol = 2τ rad³ / 3

and

rad = √( Area / 2τ )

rad = ∛( 3 Vol / 2τ )

(where π=τ/2)

In addition, I do not calculate the areas of individual faces. Those are easily obtained by dividing the total surface area S by the number of faces.

The tables are also available in PDF format.

given a = edge |
tetrahedron (4) |
cube / hexahedron (6) |
octahedron (8) |
dodecahedron (12) |
icosahedron (20) |
---|---|---|---|---|---|

a = edge |
a | a | a | a | a |

r = inradius |
a·√6/12 ≈ 0.204124⁺ · a |
a·1/2 ≈ 0.5 · a |
a·√6/6 ≈ 0.408248⁺ · a |
a·√(10·(25+11·√5))/20 ≈ 1.113516⁺ · a |
a·√(6·(7+3·√5))/12 ≈ 0.755761⁺ · a |

ρ = midradius |
a·√2/4 ≈ 0.353553⁺ · a |
a·√2/2 ≈ 0.707107¯ · a |
a·1/2 ≈ 0.5 · a |
a·(3+√5)/4 ≈ 1.309017¯ · a |
a·(1+√5)/4 ≈ 0.809017¯ · a |

R = circumradius |
a·√6/4 ≈ 0.612372⁺ · a |
a·√3/2 ≈ 0.866025⁺ · a |
a·√2/2 ≈ 0.707107¯ · a |
a·√(6·(3+√5))/4 ≈ 1.401259¯ · a |
a·√(2·(5+√5))/4 ≈ 0.951057¯ · a |

S = surface area |
a²·√3 ≈ 1.732051¯ · a² (a·∜3)² ≈ (1.316074⁺ · a)² |
a²·6 ≈ 6.0 · a² (a·√6)² ≈ (2.449490¯ · a)² |
a²·2·√3 ≈ 3.464102¯ · a² (a·∜12)² ≈ (1.861210¯ · a)² |
a²·3·√(5·(5+2·√5)) ≈ 20.645729¯ · a² (a·∜(45·(5+2·√5)))² ≈ (4.543757⁺ · a)² |
a²·5·√3 ≈ 8.660254⁺ · a² (a·∜75)² ≈ (2.942831¯ · a)² |

C = volume |
a³·√2/12 ≈ 0.117851⁺ · a³ (a·⁶√41472/12)³ ≈ (0.490280⁺ · a)³ |
a³ | a³·√2/3 ≈ 0.471405¯ · a³ (a·⁶√162/3)³ ≈ (0.778272¯ · a)³ |
a³·(15+7·√5)/4 ≈ 7.663119¯ · a³ (a·∛(16·(15+7·√5))/4)³ ≈ (1.971523⁺ · a)³ |
a³·5·(3+√5)/12 ≈ 2.181695¯ · a³ (a·∛(720·(3+√5))/12)³ ≈ (1.296974⁺ · a)³ |

given r = inradius |
tetrahedron (4) |
cube / hexahedron (6) |
octahedron (8) |
dodecahedron (12) |
icosahedron (20) |
---|---|---|---|---|---|

a = edge |
r·2·√6 ≈ 4.898979⁺ · r |
r·2 ≈ 2.0 · r |
r·√6 ≈ 2.449490¯ · r |
r·√(2·(25−11·√5)) ≈ 0.898056¯ · r |
r·√(6·(7−3·√5)) ≈ 1.323169⁺ · r |

r = inradius |
r | r | r | r | r |

ρ = midradius |
r·√3 ≈ 1.732051¯ · r |
r·√2 ≈ 1.414214¯ · r |
r·√6/2 ≈ 1.224745¯ · r |
r·√(2·(5−√5))/2 ≈ 1.175571¯ · r |
r·√(6·(3−√5))/2 ≈ 1.070466⁺ · r |

R = circumradius |
r·3 ≈ 3.0 · r |
r·√3 ≈ 1.732051¯ · r |
r·√3 ≈ 1.732051¯ · r |
r·√(3·(5−2·√5)) ≈ 1.258409¯ · r |
r·√(3·(5−2·√5)) ≈ 1.258409¯ · r |

S = surface area |
r²·24·√3 ≈ 41.569219⁺ · r² (r·2·∜108)² ≈ (6.447420¯ · r)² |
r²·24 ≈ 24.0 · r² (r·2·√6)² ≈ (4.898979⁺ · r)² |
r²·12·√3 ≈ 20.784610¯ · r² (r·2·∜27)² ≈ (4.559014⁺ · r)² |
r²·30·√(2·(65−29·√5)) ≈ 16.650873⁺ · r² (r·∜(1800·(65−29·√5)))² ≈ (4.080548⁺ · r)² |
r²·30·√(6·(47−21·√5)) ≈ 15.162168⁺ · r² (r·∜(5400·(47−21·√5)))² ≈ (3.893863¯ · r)² |

C = volume |
r³·8·√3 ≈ 13.856406⁺ · r³ (r·⁶√192)³ ≈ (2.401874¯ · r)³ |
r³·8 ≈ 8.0 · r³ (r·2)³ ≈ (2.0 · r)³ |
r³·4·√3 ≈ 6.928203⁺ · r³ (r·⁶√48)³ ≈ (1.906369¯ · r)³ |
r³·10·√(2·(65−29·√5)) ≈ 5.550291⁺ · r³ (r·⁶√(200·(65−29·√5)))³ ≈ (1.770538¯ · r)³ |
r³·10·√(6·(47−21·√5)) ≈ 5.054056⁺ · r³ (r·⁶√(600·(47−21·√5)))³ ≈ (1.716116⁺ · r)³ |

given ρ = midradius |
tetrahedron (4) |
cube / hexahedron (6) |
octahedron (8) |
dodecahedron (12) |
icosahedron (20) |
---|---|---|---|---|---|

a = edge |
ρ·2·√2 ≈ 2.828427⁺ · ρ |
ρ·√2 ≈ 1.414214¯ · ρ |
ρ·2 ≈ 2.0 · ρ |
ρ·(3−√5) ≈ 0.763932⁺ · ρ |
ρ·(√5−1) ≈ 1.236068¯ · ρ |

r = inradius |
ρ·√3/3 ≈ 0.577350⁺ · ρ |
ρ·√2/2 ≈ 0.707107¯ · ρ |
ρ·√6/3 ≈ 0.816497¯ · ρ |
ρ·√(10·(5+√5))/10 ≈ 0.850651¯ · ρ |
ρ·√(6·(3+√5))/6 ≈ 0.934172⁺ · ρ |

ρ = midradius |
ρ | ρ | ρ | ρ | ρ |

R = circumradius |
ρ·√3 ≈ 1.732051¯ · ρ |
ρ·√6/2 ≈ 1.224745¯ · ρ |
ρ·√2 ≈ 1.414214¯ · ρ |
ρ·√(6·(3−√5))/2 ≈ 1.070466⁺ · ρ |
ρ·√(2·(5−√5))/2 ≈ 1.175571¯ · ρ |

S = surface area |
ρ²·8·√3 ≈ 13.856406⁺ · ρ² (ρ·2·∜12)² ≈ (3.722419⁺ · ρ)² |
ρ²·12 ≈ 12.0 · ρ² (ρ·2·√3)² ≈ (3.464102¯ · ρ)² |
ρ²·8·√3 ≈ 13.856406⁺ · ρ² (ρ·2·∜12)² ≈ (3.722419⁺ · ρ)² |
ρ²·6·√(10·(25−11·√5)) ≈ 12.048685¯ · ρ² (ρ·∜(360·(25−11·√5)))² ≈ (3.471122¯ · ρ)² |
ρ²·10·√(6·(7−3·√5)) ≈ 13.231691¯ · ρ² (ρ·∜(600·(7−3·√5)))² ≈ (3.637539⁺ · ρ)² |

C = volume |
ρ³·8/3 ≈ 2.666667¯ · ρ³ (ρ·2·∛9/3)³ ≈ (1.386723¯ · ρ)³ |
ρ³·2·√2 ≈ 2.828427⁺ · ρ³ (ρ·√2)³ ≈ (1.414214¯ · ρ)³ |
ρ³·8·√2/3 ≈ 3.771236⁺ · ρ³ (ρ·2·⁶√162/3)³ ≈ (1.556543⁺ · ρ)³ |
ρ³·2·(3·√5−5) ≈ 3.416408¯ · ρ³ (ρ·∛(2·(3·√5−5)))³ ≈ (1.506110¯ · ρ)³ |
ρ³·10·(√5−1)/3 ≈ 4.120227¯ · ρ³ (ρ·∛(90·(√5−1))/3)³ ≈ (1.603148⁺ · ρ)³ |

given R = circumradius |
tetrahedron (4) |
cube / hexahedron (6) |
octahedron (8) |
dodecahedron (12) |
icosahedron (20) |
---|---|---|---|---|---|

a = edge |
R·2·√6/3 ≈ 1.632993⁺ · R |
R·2·√3/3 ≈ 1.154701¯ · R |
R·√2 ≈ 1.414214¯ · R |
R·√(6·(3−√5))/3 ≈ 0.713644⁺ · R |
R·√(10·(5−√5))/5 ≈ 1.051462⁺ · R |

r = inradius |
R·1/3 ≈ 0.333333⁺ · R |
R·√3/3 ≈ 0.577350⁺ · R |
R·√3/3 ≈ 0.577350⁺ · R |
R·√(15·(5+2·√5))/15 ≈ 0.794654⁺ · R |
R·√(15·(5+2·√5))/15 ≈ 0.794654⁺ · R |

ρ = midradius |
R·√3/3 ≈ 0.577350⁺ · R |
R·√6/3 ≈ 0.816497¯ · R |
R·√2/2 ≈ 0.707107¯ · R |
R·√(6·(3+√5))/6 ≈ 0.934172⁺ · R |
R·√(10·(5+√5))/10 ≈ 0.850651¯ · R |

R = circumradius |
R | R | R | R | R |

S = surface area |
R²·8·√3/3 ≈ 4.618802⁺ · R² (R·2·∜108/3)² ≈ (2.149140¯ · R)² |
R²·8 ≈ 8.0 · R² (R·2·√2)² ≈ (2.828427⁺ · R)² |
R²·4·√3 ≈ 6.928203⁺ · R² (R·2·∜3)² ≈ (2.632148⁺ · R)² |
R²·2·√(10·(5−√5)) ≈ 10.514622⁺ · R² (R·∜(40·(5−√5)))² ≈ (3.242626¯ · R)² |
R²·2·√(30·(3−√5)) ≈ 9.574541⁺ · R² (R·∜(120·(3−√5)))² ≈ (3.094276¯ · R)² |

C = volume |
R³·8·√3/27 ≈ 0.513200⁺ · R³ (R·2·⁶√3/3)³ ≈ (0.800625¯ · R)³ |
R³·8·√3/9 ≈ 1.539601¯ · R³ (R·2·⁶√27/3)³ ≈ (1.154701¯ · R)³ |
R³·4/3 ≈ 1.333333⁺ · R³ (R·∛36/3)³ ≈ (1.100642⁺ · R)³ |
R³·2·√(30·(3+√5))/9 ≈ 2.785164¯ · R³ (R·⁶√(1080·(3+√5))/3)³ ≈ (1.406966¯ · R)³ |
R³·2·√(2·(5+√5))/3 ≈ 2.536151¯ · R³ (R·⁶√(648·(5+√5))/3)³ ≈ (1.363719⁺ · R)³ |

given S = surface area |
tetrahedron (4) |
cube / hexahedron (6) |
octahedron (8) |
dodecahedron (12) |
icosahedron (20) |
---|---|---|---|---|---|

a = edge |
√S·∜27/3 ≈ 0.759836¯ · √S √(S·√3/3) ≈ √(0.577350⁺ · S) |
√S·√6/6 ≈ 0.408248⁺ · √S √(S·1/6) ≈ √(0.166667¯ · S) |
√S·∜1728/12 ≈ 0.537285¯ · √S √(S·√3/6) ≈ √(0.288675⁺ · S) |
√S·∜(225·(5−2·√5))/15 ≈ 0.220082⁺ · √S √(S·√(5−2·√5)/15) ≈ √(0.048436⁺ · S) |
√S·∜675/15 ≈ 0.339809¯ · √S √(S·√3/15) ≈ √(0.115470⁺ · S) |

r = inradius |
√S·∜12/12 ≈ 0.155101¯ · √S √(S·√3/72) ≈ √(0.024056⁺ · S) |
√S·√6/12 ≈ 0.204124⁺ · √S √(S·1/24) ≈ √(0.041667¯ · S) |
√S·∜3/6 ≈ 0.219346¯ · √S √(S·√3/36) ≈ √(0.048113¯ · S) |
√S·∜(360·(65+29·√5))/60 ≈ 0.245065⁺ · √S √(S·√(10·(65+29·√5))/600) ≈ √(0.060057¯ · S) |
√S·∜(600·(47+21·√5))/60 ≈ 0.256814⁺ · √S √(S·√(6·(47+21·√5))/360) ≈ √(0.065954¯ · S) |

ρ = midradius |
√S·∜108/12 ≈ 0.268642⁺ · √S √(S·√3/24) ≈ √(0.072169¯ · S) |
√S·√12/12 ≈ 0.288675⁺ · √S √(S·1/12) ≈ √(0.083333⁺ · S) |
√S·∜108/12 ≈ 0.268642⁺ · √S √(S·√3/24) ≈ √(0.072169¯ · S) |
√S·∜(1800·(25+11·√5))/60 ≈ 0.288091⁺ · √S √(S·√(2·(25+11·√5))/120) ≈ √(0.082997¯ · S) |
√S·∜(5400·(7+3·√5))/60 ≈ 0.274911⁺ · √S √(S·√(6·(7+3·√5))/120) ≈ √(0.075576⁺ · S) |

R = circumradius |
√S·∜12/4 ≈ 0.465302⁺ · √S √(S·√3/8) ≈ √(0.216506⁺ · S) |
√S·√2/4 ≈ 0.353553⁺ · √S √(S·1/8) ≈ √(0.125 · S) |
√S·∜27/6 ≈ 0.379918¯ · √S √(S·√3/12) ≈ √(0.144338¯ · S) |
√S·∜(200·(5+√5))/20 ≈ 0.308392⁺ · √S √(S·√(2·(5+√5))/40) ≈ √(0.095106¯ · S) |
√S·∜(27000·(3+√5))/60 ≈ 0.323177⁺ · √S √(S·√(30·(3+√5))/120) ≈ √(0.104444¯ · S) |

S = surface area |
S | S | S | S | S |

C = volume |
√S³·∜12/36 ≈ 0.051700⁺ · √S³ √(S·⁶√243/18)³ ≈ √(0.138781¯ · S)³ |
√S³·√6/36 ≈ 0.068041⁺ · √S³ √(S·1/6)³ ≈ √(0.166667¯ · S)³ |
√S³·∜3/18 ≈ 0.073115⁺ · √S³ √(S·⁶√972/18)³ ≈ √(0.174853¯ · S)³ |
√S³·∜(360·(65+29·√5))/180 ≈ 0.081688⁺ · √S³ √(S·⁶√(250·(65+29·√5))/30)³ ≈ √(0.188267¯ · S)³ |
√S³·∜(600·(47+21·√5))/180 ≈ 0.085605¯ · √S³ √(S·⁶√(303750·(47+21·√5))/90)³ ≈ √(0.194237⁺ · S)³ |

given C = volume |
tetrahedron (4) |
cube / hexahedron (6) |
octahedron (8) |
dodecahedron (12) |
icosahedron (20) |
---|---|---|---|---|---|

a = edge |
∛C·⁶√72 ≈ 2.039649¯ · ∛C ∛(C·6·√2) ≈ ∛(8.485281⁺ · C) |
∛C | ∛C·⁶√288/2 ≈ 1.284898⁺ · ∛C ∛(C·3·√2/2) ≈ ∛(2.121320⁺ · C) |
∛C·∛(25·(7·√5−15))/5 ≈ 0.507222⁺ · ∛C ∛(C·(7·√5−15)/5) ≈ ∛(0.130495⁺ · C) |
∛C·∛(75·(3−√5))/5 ≈ 0.771025⁺ · ∛C ∛(C·3·(3−√5)/5) ≈ ∛(0.458359⁺ · C) |

r = inradius |
∛C·⁶√243/6 ≈ 0.416342¯ · ∛C ∛(C·√3/24) ≈ ∛(0.072169¯ · C) |
∛C·1/2 ≈ 0.5 · ∛C ∛(C·1/8) ≈ ∛(0.125 · C) |
∛C·⁶√972/6 ≈ 0.524558¯ · ∛C ∛(C·√3/12) ≈ ∛(0.144338¯ · C) |
∛C·⁶√(250·(65+29·√5))/10 ≈ 0.564800⁺ · ∛C ∛(C·√(10·(65+29·√5))/200) ≈ ∛(0.180171¯ · C) |
∛C·⁶√(303750·(47+21·√5))/30 ≈ 0.582711⁺ · ∛C ∛(C·√(6·(47+21·√5))/120) ≈ ∛(0.197861¯ · C) |

ρ = midradius |
∛C·∛3/2 ≈ 0.721125¯ · ∛C ∛(C·3/8) ≈ ∛(0.375 · C) |
∛C·√2/2 ≈ 0.707107¯ · ∛C ∛(C·√2/4) ≈ ∛(0.353553⁺ · C) |
∛C·⁶√288/4 ≈ 0.642449⁺ · ∛C ∛(C·3·√2/16) ≈ ∛(0.265165⁺ · C) |
∛C·∛(25·(5+3·√5))/10 ≈ 0.663962⁺ · ∛C ∛(C·(5+3·√5)/40) ≈ ∛(0.292705⁺ · C) |
∛C·∛(75·(1+√5))/10 ≈ 0.623773¯ · ∛C ∛(C·3·(1+√5)/40) ≈ ∛(0.242705⁺ · C) |

R = circumradius |
∛C·⁶√15552/4 ≈ 1.249025¯ · ∛C ∛(C·9·√3/8) ≈ ∛(1.948557⁺ · C) |
∛C·√3/2 ≈ 0.866025⁺ · ∛C ∛(C·3·√3/8) ≈ ∛(0.649519⁺ · C) |
∛C·∛6/2 ≈ 0.908560⁺ · ∛C ∛(C·3/4) ≈ ∛(0.75 · C) |
∛C·⁶√(168750·(3−√5))/10 ≈ 0.710749⁺ · ∛C ∛(C·3·√(30·(3−√5))/40) ≈ ∛(0.359045⁺ · C) |
∛C·⁶√(56250·(5−√5))/10 ≈ 0.733289¯ · ∛C ∛(C·3·√(10·(5−√5))/40) ≈ ∛(0.394298⁺ · C) |

S = surface area |
∛C²·6·⁶√3 ≈ 7.205622¯ · ∛C² ∛(C·6·∜108)² ≈ ∛(19.342259¯ · C)² |
∛C²·6 ≈ 6.0 · ∛C² ∛(C·6·√6)² ≈ ∛(14.696938⁺ · C)² |
∛C²·⁶√34992 ≈ 5.719106¯ · ∛C² ∛(C·6·∜27)² ≈ ∛(13.677042⁺ · C)² |
∛C²·3·⁶√(200·(65−29·√5)) ≈ 5.311614¯ · ∛C² ∛(C·3·∜(1800·(65−29·√5)))² ≈ ∛(12.241644⁺ · C)² |
∛C²·3·⁶√(600·(47−21·√5)) ≈ 5.148349¯ · ∛C² ∛(C·3·∜(5400·(47−21·√5)))² ≈ ∛(11.681589¯ · C)² |

C = volume |
C | C | C | C | C |

For example, for :

can be partitioned as

so that

and, lastly,

Amazingly, this can be done for any non-negative integer .

Having equal sums is the same as having equal sums of first powers.

If, just for the course of this article, we accept a convention that , then the condition of having the same size may be rephrased as having equal sums of zeroth powers.

Example 1 ()can be partitioned as

so that

Example 2 ()can be partitioned as

so that

Example 3 ()can be partitioned as

so that

Example 4 ()can be partitioned as

so that

Example 5 ()can be partitioned as

so that

In order to prove the *Sum of Powers Partition Theorem*, we will prove a slightly more general, but somewhat less-interesting, result, the *Multi-partition Power Sum Theorem*.

Given the multiset (or sequence)

Define the multisets (or sequences)

The sets are sets of EVEN sums, i.e. sums of an even number of summands; the sets are sets of ODD sums.

Example 6 (initial general expansions)

Definition 2

Considering multiset union

*Proof:* (By induction on .)

Considering multiset intersection

(We may also regard this in terms of ordinary sets if the sums above are considered as formal sums, and the symbols are pairwise distinct.)

*Proof:* (By induction on .)

Lemma 3(Sum Partition — magnitude)

In fact:

*Proof:* (By induction on .)

Proposition 4 (Multi-partition Power Sum Theorem)Given any , and with and as in definition 1:

*Proof:*

Base case: :

The only to consider is . But .

Inductive case.

Suppose the theorem holds for , and consider the case .

Part 1. Consider .

The left summands are equal by inductive hypothesis, so we must show equality of the right summands.

Part 2. Consider .

From parts 1 and 2, we have, respectively,

Thus

which completes the proof.

Note that the (swap) step in this proof, above, is required as, generally,

Example 7 (particular power expansions)

Definition 3Let

And now … the main result:

Corollary 5 (Partition Power Sum Theorem)

*Proof:*

Let . Let for .

Let , .

By proposition 4, the sets and satisfy

We have

By lemmas 1, 2 and 3, , form a partition of . That is, and form a partition of , which completes the proof.

The sets are sets of so-called EVIL numbers; the sets are sets of ODIOUS numbers.

Conjecture 6The partition in (2) is unique (up to exchange of and ).

There is empirical evidence for this. A brute-force search by computer finds only (and exactly) one solution for cases up to .

To disprove the conjecture, only a single counterexample need be found.

A proof of the conjecture might be found by a *descent* approach, perhaps.

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QiX is a library for Albert Gräf’s Q programming language adding support for univariate polynomials.

There is full documentation available.

]]>Q+Q is a library for Albert Gräf’s Q programming language adding support for the rational numbers, ℚ.

There is full documentation available.

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