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The human heart: Application of the golden ratio and angle. Michael Y. Henein and the Golden Ratio Collaborators: Ying Zhao, Rachel Nicoll , Lin Sun,Ashraf W. Khir , Karl Franklin, Per Lindqvist. Reporter:Shu-Hsiu,Yu Date: 2012/06/19. Volume 150, Issue 3, 4 August 2011, Pages 239–242.

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the human heart application of the golden ratio and angle

The human heart: Application of the golden ratio and angle

Michael Y. Henein and the Golden Ratio Collaborators: Ying Zhao, Rachel Nicoll, Lin Sun,Ashraf W. Khir, Karl Franklin, Per Lindqvist


Date: 2012/06/19

Volume 150, Issue 3, 4 August 2011, Pages 239–242

  • The golden ratio (also known as the golden mean or divine proportion) is so called when on a line consisting of a longer length (A) plus a shorter length (B), the proportion (A+B)/A equals A/B.


It is expressed as an irrational mathematical constant phi (φ) of approximately1.618.

  • This proportion is said to be the mostaesthetically pleasing and many artists andarchitects have incorporated it into their work.
  • The golden ratio has also been found within the proportions of the human body, including limbs, facial features, teeth and the DNA molecule, and quantum field theory.

Allied to this is the golden angle, calculated by sectioning the circumference of a circle (c) into two arcs, such that the ratio of the length of the larger arc (a) to the length of the smaller arc (b) is the same as the ratio of the full circumference to the length of the larger arc (i.e. c/a = a/b).


The golden angle is the angle subtended by the smaller arc (b) and measures 137.5°. The golden ratio decimal (0.618) equates to an angle of 222.5°, the reverse of the golden angle, with the sum adding up to 360°.

  • It has been found among branches and leaves on plant stems.


Authors hypothesised that since the golden ratio and angle are so deeply rooted in nature, they must have some correspondence in the healthy heart structure and function.

  • Accordingly, we studied the heart dimensions and function using 2D or 3D cardiac ultrasound imaging technology at end-diastole, the resting phase of the heart cycle or CT scanning. Individual study data were compared using the unpaired t test.
an echocardiographic example of normal 4 chambers of the heart
An echocardiographic example of normal 4 chambers of the heart
  • To assess the effect of ethnicity on the left ventricle, the most important heart pump, authors took vertical and transverse cardiac measurements from 30 normal Swedish adults and compared these with measurements from 30 healthy matched Chinese.

Authors found that Chinese left ventricle dimensions were smaller by 8 ± 1 mm and 5 ± 0.5 mm, compared to Swedes but the golden ratio was maintained at approximately 1.618.

  • Authors’ findings suggest that it is the dimensions ratio that is important for healthy functioning and not the absolute dimensions.
differences between the two heart failure groups and normals
Differences between the two heart failure groups and normals
  • To assess whether the ratio of these dimensions differs in heart failure, authors tested them in 40 Swedish patients; 20 with mild heart failure and 20 with end-stage heart failure.
  • In mild heart failure, the left ventricular ratio was reduced (p < 0.001) but the right ventricular ratio was increased due to the enlarged left ventricle encroaching on the right ventricular dimensions (p < 0.001); overall the cardiac ratio remained at around 1.64.

However, in end-stage heart failure, the left ventricular ratio was significantly reduced (p < 0.01) and this distorted the cardiac ratio to around 1.4 (p < 0.005), consistent with pathological remodeling resulting in a more spherical shape to the heart.

  • These patients had 50% mortality over the course of 36 months' follow up, compared with 100% survival in the mild group.

Consequently, it appears that as long as the cardiac ratio is maintained at approximately 1.618, survival is likely but a significant deviation from the golden ratio is associated with a poor outcome.

to e valuate the left ventricular inlet length and width
Toevaluate the left ventricular inlet length and width
  • Authors evaluated the left ventricular inlet length and width (mitral annulus dimensions) in 15 normal individuals and 12 age matched patients with dilated cardiomyopathy and functional mitral regurgitation.
  • In normals, the relative mitral annulus circumference length and width proved to be 6.8 ± 0.8 cm and 4.2 ± 0.8 cm respectively, giving a ratio of 1.62, compared with the dilated cardiomyopathy patients, whose dimensions were 8.3 ± 1.8 cm and 5.8 ± 1.1 cm respectively, giving a ratio of 1.42 (p < 0.001).

Authors have also previously demonstrated that the golden ratio is seen in normal foetal myocardial function development, where the heart muscle diastolic function matures at a rate of 1.6 mm/s per week between 20 and 40 weeks of age, while a healthy adult heart muscle loses velocity of 1.6 cm/s each decade between age 20 and 80 years, indicating a close relationship between ageing and ventricular myocardial function.


Finally, others have shown that the end product of normal cardiac function, the blood pressure ratio of 120/80 mmHg, also approximates to the golden ratio of 1.618 and carries significant prognostic value.


Additionally, myocardial muscle fibre orientation takes a helical shape, with the coils of each segment of the apical loop being of different lengths and having a harmonic proportion which conforms to the golden ratio.

  • When these myocardial fibres become horizontal or transverse, as is the case in heart failure, they weaken overall ventricular function.
to assess the anatomical angle
To assess the anatomical angle
  • Authors assessed the anatomical angle between the right ventricular inlet axis and outflow tract axis in 16 healthy subjects and 19 patients with severe pulmonary hypertension and right heart failure. We found that the normal angle between the axes is approximately 43 ± 3°.
  • The angle between the outflow tract axis and continuation of the inflow tract axis is therefore 138 ± 4°, which is approximately 137.5°.

This angle increased significantly to 160 ± 4° (p > 0.001) in right heart failure patients, as the right ventricle became cylindrical and lost its native shape, reducing the inflow-outflow tract angle to approximately 20° from 43° (unpublished).


Hence it can be seen that the complex anatomy of the right ventricle, as well as its myocardial fibre architecture, support the unique role of this angle in preserving the peristaltic circulation inside the right ventricle.

  • With severe disease, right ventricular cavity enlargement and remodelling combine to produce loss of the normal golden angle, contributing to the intractable right ventricular pump decompensation.
to measure the angle
To measure the angle
  • Authors then measured the angle between the midluminal axes of the pulmonary trunk and the proximal ascending aorta in the same 16 healthy subjects.
  • This angle approximated to a mean of 39.5 ± 3.6°, making the angle between the pulmonary trunk and the continuation of the ascending aorta approximately 139 ± 3°, which corresponds with the golden angle of 137.5° (unpublished).

Normal systolic pressure in the aorta is almost six times that in the pulmonary artery, hence the need for a pressure recovery space between the two arteries to avoid pressure transmission across their thin walls. In hypertension, the pressure difference between the aorta and the pulmonary artery could increase to 10 times the norm.

  • Had the aorta and pulmonary artery been closely parallel, this could potentially have created a significant afterload to the right ventricle, which is avoided by the space provided by the angle between the axes.

In addition to these two great arteries, the same angle seems to be adopted by the central and peripheral arterial tree; early anatomical studies demonstrated the branching pattern of the coronary and peripheral arteries to be compatible with the golden angle.

  • It appears that the golden angle is not only important for coronary vasculature packing but also for optimising the branching pattern so that perfusion of the myocardial bed is maximised.
  • Authors’ findings show that the overall cardiac and ventricular dimensions in a normal heart are consistent with an approximation of the golden ratio of 1.618 and the golden angle of 137.5°, representing optimum pump structure and function efficiency.
  • These findings are not only of philosophical interest but may have a significant anatomical, functional and prognostic value.

Application of these ratios and angles in our daily clinical practice may lead to development of simple and reliable markers of early deviation from normality, which could be treated before irreversible alteration in cardiac structure and function develops.