The question of generalising circle packing to three dimensions was asked in [65677](https://mathoverflow.net/questions/65677/). There is a clear consensus that there is no obvious three dimensional version of circle packing.


However I have seen a comment that circle packing on surfaces and Ricci flow on surfaces are related. The circle packing here is an extension of circle packing to include intersection angles between the circles with a particular choice for these angles. My initial question is to ask for an explanation of this.


My real question should now be apparent. There is an extension of Ricci flow to three dimensions: so is there some version of circle packing in three dimensions which can be interpreted as a combinatorial version of Ricci flow?



Is there a combinatorial analogue of Ricci flow?
Actually, there are a number of references by Ben Chow, Feng Luo, and D. Glickenstein on this subject, mostly in two dimensions. Glickenstein's work (Glickenstein was a student of Ben Chow's) is more three-dimensional. Some relevant references are below. The curvature flow approach distinct from the even more popular variational approach (though the two approaches intersect nontrivially).


[MR0127372](https://mathscinet.ams.org/mathscinet-getitem?mr=0127372) (23 #B418) 
Regge, T.
General relativity without coordinates. (Italian summary) 
Nuovo Cimento (10) 19 1961 558–571. 


[MR1393382](https://mathscinet.ams.org/mathscinet-getitem?mr=1393382) (97k:52022) 
Cooper, Daryl(1-UCSB); Rivin, Igor(4-WARW-MI)
Combinatorial scalar curvature and rigidity of ball packings. 
Math. Res. Lett. 3 (1996), no. 1, 51–60. 


[MR2136536](https://mathscinet.ams.org/mathscinet-getitem?mr=2136536) (2006a:53081) Glickenstein, David A maximum principle for combinatorial Yamabe flow. Topology 44 (2005), no. 4, 809–825. (Reviewer: Igor Rivin), 53C44 (52C15) 


[MR2136535](https://mathscinet.ams.org/mathscinet-getitem?mr=2136535) (2005k:53108) Glickenstein, David A combinatorial Yamabe flow in three dimensions. Topology 44 (2005), no. 4, 791–808. (Reviewer: Igor Rivin), 53C44 (52C15) 


[MR2100762](https://mathscinet.ams.org/mathscinet-getitem?mr=2100762) (2005m:53122) Luo, Feng Combinatorial Yamabe flow on surfaces. Commun. Contemp. Math. 6 (2004), no. 5, 765–780. (Reviewer: Igor Rivin), 53C44 (53C21) 


[MR2015261](https://mathscinet.ams.org/mathscinet-getitem?mr=2015261) (2005a:53106) Chow, Bennett; Luo, Feng Combinatorial Ricci flows on surfaces. J. Differential Geom. 63 (2003), no. 1, 97–129. (Reviewer: Igor Rivin), 53C44 


[arXiv:1010.4070](https://arxiv.org/abs/1010.4070) [pdf, ps, other]
Discrete Laplace-Beltrami Operator Determines Discrete Riemannian Metric
Xianfeng David Gu, Ren Guo, Feng Luo, Wei Zeng


[arXiv:1005.4648](https://arxiv.org/abs/1005.4648) [pdf, other]
Computing Quasiconformal Maps on Riemann surfaces using Discrete Curvature Flow
W. Zeng, L.M. Lui, F. Luo, J.S. Liu T.F. Chan, S.T. Yau, X.F. Gu


