Heinrich HERTZ's Contact Law and NPC Passage by the HIV Capsid: Contact Mechanic Is Putative Relevant by the Twelve HIV Pentagons: The FIBONACCI number 5 at the Narrow End, and the LUCAS Number 7 at the Broad End and the FIBONACCI Numbers 3, 8, and 34 in the NPC

Heinrich HERTZ's Contact Law and NPC Passage by the HIV Capsid: Contact Mechanic Is Putative Relevant by the Twelve HIV Pentagons: The FIBONACCI number 5 at the Narrow End, and the LUCAS Number 7 at the Broad End and the FIBONACCI Numbers 3, 8, and 34 in the NPC

HIV-1 obeys Euler’s polyhedron formula with 12 CA pentamers capping the ends of the cone: 5 pentamers at the narrow end, 7 at the broad end (see Figure 4 etc. from “Passage of theHIV capsid cracks the nuclear pore“ by Kreysing, Jan Philipp et al., Cell, Volume 188, 2025, Issue4, 930 - 943.e21).
This figure shows the fifth FIBONACCI number F(5) = 5 at the narrow end, and the fourth LUCAS Number L(4) = 7 at the broad end of the near ellipsoid cone.
Thus, even the substructure of the HIV capsid is in accordance with GEIER’s equations allowing FIBONACCI and LUCAS numbers. As L(n) = √5F(n) this is with 5=F(5) and √5 in direct relation to Daniel SCHECHTMAN’s quasicrystals by 5√5 = 5^(3/2) = 11.180. .. ≈ L(5) (and L(6)/ √5 = 18/√5 = 8.0498 . .. ≈ F(6) ; or with L(7)=13 by 29 ⋅1/√5 = 12.969...; the LUCAS number 29 is less impressive except that it’s a prime and a quadraticnumber based on 5 and 2 nota bene giving 22+52√5 , and it is narrowing the interval for the 12 pentamers).
This is another hint for the relevance of GEIER’s equations based on physical quanta such as the elementary charge e and 2ħ or √2ℏ. Furthermore, the exponent 3/2 brings in Heinrich HERTZ’s contact law of contact mechanics, of mechanic quadrupoles (balls in Isaak NEWTON’s Cradle), and of gravitational quadrupoles as described by our group while studying the bifilar pendulum, and NEWTON’s Cradle. This suggests that the HIV capsid is in an (near) optimal equilibrium of electrostatic forces and the gravitational force. Therefore, the HIV capsid and related structures should be considered as a (putative) appropriate model to study quantum gravitation in physical experiments and in computer simulations.

It's of substantial interest that the NPC octogonal structure
1. with 3 rings, 8 edges, and 34 proteins maximizes stability by the three FIBONACCI numbers 3, 8, and 34 while
2. allowing flexibility, oscillaton, or cracking by not following "GEIER's n x 6° rule" by 360°/8=45°.

This demonstrates the relevance of GEIER's equation in virology and biochemistry very impressive.

Yours Stefan Geier, Haidholzen

(PDF) The HIV-1 Capsid, FIBONACCI Numbers, LUCAS Numbers, Geoffrey HINTON's Backpropagation, and GEIER's Equation Based on the Elementary Charge e and the Doubled Reduced PLANCK Constant 2ħ (HIV Capsid and Quantum Physics Part 2.0). Available from: https://www.researchgate.net/.../398446260_The_HIV-1... [accessed Dec 13 2025].

cell.com

Passage of the HIV capsid cracks the nuclear pore

Cone-shaped HIV-1 capsids with largely intact lattices accumulate at and pass through nuclear pores in primary human macrophages. The nuclear pores can crack open upon passage of the HIV-1 capsid, possibly to relieve a steric blockage of capsids inside pores.