A version of Murray's Law was articulated by Leonardo Da Vinci https://phys.org/news/2012-01-leonardo-da-vinci-tree.html On Fri, Oct 12, 2018, 05:00 <math-fun-request@mailman.xmission.com> wrote:

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Today's Topics:

1. Re: Simple model for branching network flows (Henry Baker) 2. Dice rotation games (James Propp) 3. Re: Dice rotation games (Mike Stay)

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Message: 1 Date: Wed, 10 Oct 2018 13:12:56 -0700 From: Henry Baker <hbaker1@pipeline.com> To: math-fun@mailman.xmission.com Subject: Re: [math-fun] Simple model for branching network flows Message-ID: <E1gAKrS-000DlO-T7@elasmtp-mealy.atl.sa.earthlink.net> Content-Type: text/plain; charset="us-ascii"

As I thought, this is an extremely well-studied & modelled area. The following is for incompressible fluid (blood) flow.

For example, Murray's Law for a bifurcation:

r^alpha = r1^alpha + r2^alpha

where r is the radius of the parent node, and r1,r2 are the radii of the child nodes. alpha is a constant >=2.

alpha=2 => sum of cross sectional areas of the children = cross sectional area of the parent.

alpha=3 => minimizes work for laminar flow.

alpha=7/3 => minimizes work for turbulent flow.

https://en.wikipedia.org/wiki/Murray%27s_law

Also,

Adam, John A. "Blood Vessel Branching: Beyond the Standard Calculus Problem". Math. Mag. 84 (2011), 196-207.

Adam calculates the height of the binary tree as around k ~ 30, so we don't need a 64-bit machine!

Adam claims (sum area of children)/(area of parent) ~ 1.26 ~ 2^(1/3).

https://en.wikipedia.org/wiki/Hemodynamics

Cross-sectional area of the aorta (root) ~ 3-5 cm^2 velocity ~ 40 cm/s. Total area of the capillaries ~ 4500-6000 cm^2 velocity ~ 0.03 cm/s. i.e., total area ratio ~ 1000 = 10^3.

--- So if rc is the radius of each child (exactly 2 children), and if rp is the radius of the parent, then

(2*pi*rc^2)/(pi*rp^2) = 2^(1/3)

So, rc/rp ~ 2^(-4/9) ~ 0.735

But Murray's Law gives

rp^alpha = 2*rc^alpha, or

1/2 = (rc/rp)^alpha, or

rc/rp = (1/2)^(1/alpha)

Putting these equations together yields a Murray-type law with alpha = 9/4 = 2.25, i.e., rp^(9/4) = 2*rc^(9/4).

The next thing to do is to estimate the flow resistance for each level 1-30, which are all in *series*.

At 04:21 PM 10/9/2018, Henry Baker wrote:

...

I'm interested in a balanced binary tree of air flow; the flow comes in at the root, and goes through k levels of binary branching, yielding 2^k outputs.

I'm assuming that air flow resistance varies superlinearly with the flow -- i.e., trying to push 2x amount of air should face a resistance which is >2x -- perhaps 4x or 8x. Let's assume that whatever exponent is used, perhaps 2<alpha<3, the same alpha occurs at each level of the tree, so that the scaling is simple fractal.

For the moment, let's assume that these pipes are cylindrical, and that the joint where one pipe becomes two is designed so that the primary resistance isn't the joint itself, but merely the sum of the resistances of the branches. There is also a slight "enlargement factor" at each branch, whereby the area of the branches at level k+1 is slightly larger than the area of the branches at level k.

My intuition is that as the flow increases due to increased pressure at the root, the maximum resistance occurs at smaller and smaller k's -- i.e., closer to the root. Equivalently, as the pressure at the root decreases, the maximum resistance occurs at larger & larger k's -- i.e. further from the root.

Is this intuition correct?

Any references or links?

Next question: what happens if the fluid is *incompressible* -- e.g., water?

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Message: 2 Date: Thu, 11 Oct 2018 10:21:26 -0400 From: James Propp <jamespropp@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: [math-fun] Dice rotation games Message-ID: <CA+G9J-fUiqQOtvnORWvxQ8Sa+Qsv=Fc5U0F1PJNitECu8= o1Tg@mail.gmail.com> Content-Type: text/plain; charset="UTF-8"

Colm Mulcahy's Hamilternion Card Games page

http://www.mathsireland.ie/hamilternion

made me wonder if there are any games that (a) hinge on the fact that 3D rotation is noncommutative and (b) require only ordinary dice.

One idea I like is that we can have one die act on another: if the first die has its m-face facing upward (for m between 1 and 6), we rotate the second die ninety degrees clockwise around the face that shows an m.

Note that the operations of the form "rotate this die ninety degrees clockwise around the face that shows an m" generate a group. It's not the usual way we think about rotation groups, since the specified axes are carried along by the object. A fun way to see that this is still associative is to imagine that the operation is "hold the object still and rotate the rest of the universe ninety degrees counterclockwise around the face that shows an m".

To make a game that's fun to play, we'd probably want to incorporate a random element. But hey, we have dice!

To make a game that's fun to play, we'd probably want players to be able to score "points". But hey, we have dice!

Anyone want to suggest a game along these lines that's got elegantly simple rules and is fun to play? Maybe I'll try out a few such games at the upcoming Celebration of Mind.

Jim Propp

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Message: 3 Date: Thu, 11 Oct 2018 08:56:08 -0600 From: Mike Stay <metaweta@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Dice rotation games Message-ID: <CAKQgqTb5EDX_sH++= eHTeRQSYrCG9Tk16xQy3x522fjANT6F9g@mail.gmail.com> Content-Type: text/plain; charset="UTF-8"

I think such a game would be possible in the style of the various games involving the orientation of rolled pig knuckles relative to each other. The modern version my family has is called "Pass the Pigs" (https://en.wikipedia.org/wiki/Pass_the_Pigs). In that game, there's a score for each configuration, but one of the more common configurations causes you to lose all your accumulated points for that turn, so you're constantly faced with the decision of whether to roll again or to pass the dice to the next player and keep the points you've accumulated so far. On Thu, Oct 11, 2018 at 8:22 AM James Propp <jamespropp@gmail.com> wrote:

...

Colm Mulcahy's Hamilternion Card Games page

http://www.mathsireland.ie/hamilternion

made me wonder if there are any games that (a) hinge on the fact that 3D rotation is noncommutative and (b) require only ordinary dice.

One idea I like is that we can have one die act on another: if the first die has its m-face facing upward (for m between 1 and 6), we rotate the second die ninety degrees clockwise around the face that shows an m.

Note that the operations of the form "rotate this die ninety degrees clockwise around the face that shows an m" generate a group. It's not the usual way we think about rotation groups, since the specified axes are carried along by the object. A fun way to see that this is still associative is to imagine that the operation is "hold the object still

and

...

rotate the rest of the universe ninety degrees counterclockwise around the face that shows an m".

To make a game that's fun to play, we'd probably want to incorporate a random element. But hey, we have dice!

To make a game that's fun to play, we'd probably want players to be able to score "points". But hey, we have dice!

Anyone want to suggest a game along these lines that's got elegantly simple rules and is fun to play? Maybe I'll try out a few such games at the upcoming Celebration of Mind.

Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

-- Mike Stay - metaweta@gmail.com http://reperiendi.wordpress.com

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