very funny video
I remember watching that a few years back. The test is only a go-nogo test. In the video, the bishop easily slides between the king and queen on a 55mm board, but it will also pass the same test on any board larger than 55mm as well. So which board is the right one? And what about sizing the remaining pieces (knight, rook, and pawns)? Shouldn't they be considered as well?
There is no rigid rule. But there are general useful rules like this with royal pair and bishop "dance" 
. Indeed, for a harmonious proportion several rules must be implemented, not just one. Others very useful general rules that I realy like it for a long time ago, which have been already exposed here even by others, are:
-for those who like it more air space-four pawns must fit about on a square.
- or for those who like it more tighter space- two pawns must fit about on the diagonal of the square
-King base diameter must be about 75%-80% from the square.
-King hight must be slightly smaller than two squares
-There may be other useful rules that do not contradict the above rules
If all this general rules match simultaneously with chess pices and the board, it will be a good proportion chess set and a pleasant play game.
loubalch's sizing chart looks optimal to me. Seems to track well with the USCF sizing guidelines, which I realize are not universal. Just my $.02.
For those who are not so inclined to "plug and chug" through the numbers, I offer the following tables to summarize the recommended king and pawn dimensions for various size chess boards.
For those living in most of the world, here's the table in centimeters (cm).
NOTE: The deviation is a measurement of how close (or not so close) a chess set is from the ideal ratios as proposed in the formulas. The closer the number is to zero, the close the set is to the ideal. A number of 5 or less indicates a very well balanced chess set.
For those who are not so inclined to "plug and chug" through the numbers, I offer the following tables to summarize the recommended king and pawn dimensions for various size chess boards.
For those living in most of the world, here's the table in centimeters (cm).
NOTE: The deviation is a measurement of how close (or not so close) a chess set is from the ideal ratios as proposed in the formulas. The closer the number is to zero, the close the set is to the ideal. A number of 5 or less indicates a very well balanced chess set.
Thank you very much for sharing!
The “curious relationship” reported in this topic arises from the similarity of triangles: the king's base is to the side of the square just as the pawn's base is to the king's base, which results in the formula B² = bS.
B: King's Base; b: Pawn base; S: Side of the square
This formula has no practical application for sizing the king and pawn bases because it is that of a parabola: the king's base grows exponentially in relation to the side of the square.
With x being the pawn scale factor, we have b = xS.
Substituting into the formula, we have B = S √x. Therefore, the king's scale factor is √x.
The reported “spooky part” (the king’s scale factor is equal to the square root of the pawn’s scale factor), falls into obvious identity: √x=√x. Thus, given any factor for the pawn, its square root will always be the king's factor.
In the topic, for b=58.6%S (two pawns diagonally on the same square), we have B=76.5%S, a value that was attributed to the king's scale factor, adopted as an average among some recommendations and which satisfies the aforementioned formula.
This base does not take into account that there must be a minimum distance between a piece that moves diagonally between two others with the same base, to avoid creating visual blocks, given by the formula S = (B+d) √2.
In short, and as always, what really matters is YOUR personal preference.
When sliding a piece on the board, diagonally, between two others with the same base, we have the following formula: S = (B+d) √2
S: Side of the square; B: Base of Kings and Queens; d: distance between bases
Instead of just the kings having, individually, the largest bases, we adopted the same base, B, for kings and queens. We then have four pieces in the game with the largest base, which better balances the relationship between the number of pieces per base.
Thus, the king and queen have different heights, but share the same royal base.
This is what already happens in several chess games, which have only three bases: the largest for kings and queens, the smallest for pawns and the average for bishops, knights and rooks.
For the formula above, it would be a queen between the opposing king and queen. By setting a minimum distance d between these pieces, all the others will move freely.
During the game, we sometimes slide the pieces over the board, sometimes we lift the piece from its initial position, describe an arc with it and lower it to the destination position. But the latter is just the movement that occurs after the decision is made, after the reasoning is completed.
During reasoning, we do not make arcs in the air with the pieces and we do not project their movements onto images of a cramped board full of obstacles.
More than a mathematical parameter, a reasonable distance between the pieces has positive effects on thinking, as it is precisely this distance that, by suppressing physical obstacles, makes reasoning more fluid and faster, enabling the perception of a clear visual path for displacements the pieces on the surface of the board.
That's why you see so many chess games around the world adopting a classic proportion, B=(2/3)S, which preserves a necessary minimum distance between the pieces.
This proportion, B=66.7%S (Kings and Queens base), perfectly meets Fide's recommendation, of 4 Pawns in the same square, b=50%S (Pawns base), as they are relatively close, differing by just 16.67%.
For more distant proportions (B=76.5%S; b=50%S), the difference jumps to 26.5% and the pawns become solitary, segregated, reduced in their bases. In this case, aiming for a better balance, there are proposals for b=58.6% S (two pawns diagonally on the same square).
However, for B=76.5%S, the calculation of the minimum distance d results in a negative value, which gives rise to collisions and the possible dropping of pieces during the game.
In the limit, for d=0, resulting in B=70.7%S, the pieces touch each other, leaving no space for the pieces to slide freely on the board. For any proportion above this, visual blocks are created that impact reasoning.
If you want – due to personal preference – for there to be a minimum value for the spacing d, consider sizing B in the range between 5/8 to a maximum of 2/3 of S, and b=50%S.
You may be over-thinking it.
Practical sizing:
A – Height of the king
B – Base of kings and queens
a – pawn height
b – pawn base
S – Side of the square
A=2a (height of the king = twice the height of the pawn)
S=2b (base of the pawn = half the side of the square)
Similarity of triangles: the height of the king is to the side of the square as the height of the pawn is to its base:
A/S = a/b
Tower height = side of the square.
Bishop, knight and rook bases = average between king and pawn bases.
S = (B+d) √2
Where d is the distance that makes it possible to slide a piece between two others with the same base arranged diagonally (a queen between the opponent's king and queen).
B = (2/3) S (king's base = two-thirds of the side of the square)
Classic, usual proportion, which preserves a minimum distance d.
In the limit, for d=0, B=70.7%S. For any proportion above this, visual blocks are created that can impact reasoning.
On a crowded board, pieces do not slide freely, they are easily knocked over and the visual paths of movement are not clearly visible.
A = (7/4) S (height of the king = seven quarters of the side of the square)
Also common proportion, with small variations, compatible with Fide specifications (H=95 and S=55 mm) and with sets used in championships.
As for the other pieces, note that the average referential inclination from the king to the pawn is around 10°, but avoid standardizing the linearly decreasing heights.
The King's inclination towards the Queen (royal couple who share the same base), for example, must be gentler than the Queen's inclination towards the bishop. From the bishop to the knight, as well as from the rook to the pawn, the inclinations will also be smoother.
In short, what really matters is YOUR personal preference.
very funny video
https://www.youtube.com/watch?v=zJeVid62NZo