Heart size

Not all hearts are of the same size. Hearts can be tiny (like jewels) or can be monumental. When looking at a picture of a heart, it's hard to determine its size.  Hearts can vary in stability too. The picture below shows figures of the same height (6 units where the unit can be anything, e.g. meter). The heart on the right shows a rectangle of length/width equal to 2. Hence, we define this ratio r=2. The ratio of the two other hearts are calculated similarly. In general, the ratio of a heart can be any value between 1 and 2. The heart with best stability has a ratio of 2.  
This ratio "r" is one way to define the shape of a heart". Another way is to start from the rectangle shown in the figure below. Here a rectangle of ratio "x=0.5" is used but it can have any value. This rectangle is used to build the heart of ratio "r=5/3" as shown in the middle. On the far right, one can see that the rectangle of shape "x" also ​appears in a different area of the heart.
So, a heart  can be uniquely defined using the following two parameters:
  • A scaling factor "s" that can take any real positive value. The larger "s" is, the larger the heart is.
  • A parameter "x" that can take any real positive value. 
Given a value of "x", one can demonstrate that the value of "r" is:  r = (x+2)/(x+1). Inversely, given a value of "r" (between 1 and 2), the value of "x" is: x= (2-r)/(r-1). For example:
  • when "x=0", "r=2" 
  • when "x=1", "r=3/2"
  • when "x=3/2" ,"r=7/5" 
  • "x" is infinitely large ,"r=1"
Some relations between "x" and "r" are quite unique. For instance,
  • when "x=sqrt(2)" (i.e. the square root of 2, that is equal to 1,4142...), then "r=sqrt(2)";
  • when "r" is the Golden ratio, i.e. "1,618...",  then "x" is the inverse of the Golden ratio;
  • when "r" is the inverse of the Golden ratio", then "x" is the Golden ratio.
A heart whose "r" is the Golden ratio is shown here below. 

Square root of 2 and Golden number

 Here below, three very special hearts are being shown. In the middle, the one of ratio "r=3/2". To its left, one of ratio "r=7/5" that is close to the square root of 2 (1.414...). To its right, one of ratio "r=8/5" that is close to the golden number (1.618...). 
Note that rectangles of the following ratio are very popular in Origami:
  • Sqrt(2), this is the so-called Silver rectangle, and the format of European standard A4;
  • Golden number (i.e. 1.618...)", this is the Golden rectangle;
  • Sqrt(3), the rectangle of this ratio is called the Bronze rectangle;
  • 2, this is the so-called double-square rectangle.
All these enable the creation of interesting heart modules.

Hearts and fractals

 Modular hearts can be assembled together to create various fractals. In order to demonstrate this, we will make use of Golden hearts. Note that similar fractals could be created by using hearts of ratio different from the Golden number.
A first fractal to create is shown in the figure below. The sculpture consists of hearts here on 5 levels. Hence, we  name it S(5). Please note that the heights of the hearts double from one level to the next. Let H(5) define the height of the bottom heart of S(5). In order to create S(6), one can do it as follow. Take two sculptures S(5) and put them next to each other on top of one modular heart of height H(6)=2xH(5). We can generalize this by saying that, at level n, H(n)= 2xH(n-1).
Another sculpture can be created by using instead: H(n)= sqrt(2)xH(n-1). By doing so, it is now the area of the heart that doubles at each level. The figure here below shows in descending order the progression of heart sizes in that case. 
A sculpture in the shape of a spiral can be created by attaching such hearts with a rotation of 45 degrees at each step. The result is shown here below. 
Another example of fractal is shown here below. In this case, a sculpure S(n) is created by assembling on a heart two sculptures S(n-1). These two are first rotated. A clockwise rotation of 45 degrees is applied to the left heart and a counterclockwise one is applied to the right heart. The two sculptures can then be attached to a heart of height H(n)= sqrt(2)xH(n-1) in order to obtain S(n). ​ When n=1, the sculpture is simply one heart. When n=2, the sculpture consists of one heart with two hearts placed on top with a 45 degree angle as explained before. Etc.
The figure here below shows the result for n=3. It could represent someone, with the 2 parents and the 4 grandparents all together in one sculpture S(3). 
Thanks to the geometric properties of the modular heart, one can simply create family trees by piling up the generations on top of each other, either in ascending order or in descending order. I particularly like the one where the new generation sits on top the previous ones as shown here. Each member of the family tree is a heart.
In order to better understand the scaling of this particular sculpture, here below is a view from the top of how hearts are to be placed. A well-know H-tree is used to show the positionning of 63 hearts representing a family tree of 6 generations of hearts (someone+2 parents +4 grandparents+8 great-grandparents+16 great-great-grandparents+32 great-great-great-grandparents