I think Pi (the relation between the circumference of a circle and it's diameter) has different values in non-Euclidean geometries. I am interested in what is invariant across all geometries. I wonder if the golden ratio may be invariant across spherical and/or hyperbolic geometries.

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    $\begingroup$ The ratio of the circumference to the radius in, say, hyperbolic geometry is not a constant. see, e.g., this As to the golden ratio, well...how are you defining it? $\endgroup$ – lulu Dec 10 '20 at 23:24
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    $\begingroup$ Pi and the golden ratio both have many (equivalent) mathematical definitions, and not just the simple geometric one(s). The ratio of a circle's cirumference to its radius may vary, but $\pi$ does not. When you ask whether 'the golden ratio' is invariant you have to be very specific about what quantity you're asking about. $\endgroup$ – Steven Stadnicki Dec 10 '20 at 23:26
  • $\begingroup$ I am thinking of the geometric "picture" of phi. Divide a line segment at a point on the segment such that the ratio of the smaller sub-segment to the larger sub-segment is equal to the ratio of the larger sub-segment to the original line segment. This seems simpler than a circle, which needs parameters (center and radius) so there are at least two parameters necessary to make a circle; but I think there are no parameters necessary for the golden ration other than the line segment (i.e. any line segment.) $\endgroup$ – Joe Cash Dec 10 '20 at 23:30
  • $\begingroup$ @Steven Stadnicki: I think Pi does vary. It is different in spherical geometries, and is not even invariant within a fixed (or given) spherical geometry (I think.) The fact that the definitions are equivalent suggest there is some kind of objective "thing" underlying all the concepts. That's not to say the "thing" is material or physical; but its' defining properties are invariant. I wrote another comment above where I discuss what I mean by the golden ratio. $\endgroup$ – Joe Cash Dec 10 '20 at 23:36
  • $\begingroup$ maa.org/external_archive/devlin/devlin_05_07.html $\endgroup$ – Will Jagy Dec 11 '20 at 0:03

Insofar as the Golden Ratio $\phi$ is defined as a number satisfying $\phi^2=\phi+1$ (this is equivalent to the extreme/mean ratio division description), the ratio belongs to any geometry where (loosely speaking) lines are effectively rulers with continuous and uniform real-number labels for points. The existence of $\phi$ in this sense is no more or less remarkable than the existence of $\frac12$. Even $\pi$ as a number exists in such contexts: it lives just past $3$ on the ruler. In a "uniform" geometry (such as the Euclidean, Spherical, and Hyperbolic planes), the rulers don't change from here to there, so the "invariance" of $\phi$ and $\frac12$ and $\pi$ as numbers is baked-in: every segment has a point dividing it in the ratio $1:x$ for any number $x$. As a matter of numerics, there's really nothing to discuss.

Where things get interesting is investigating whether a value like $\phi$ "arises" from the geometry. For instance, one way $\phi$ arises in Euclidean geometry is via the regular pentagram:

enter image description here

Sides $\overline{AB}$ and $\overline{PQ}$ meet at a point $K$ that divides each of those sides into the ratio $1:\phi$. This construction is "invariant" in Euclidean geometry because all regular pentagrams are similar, and corresponding ratios are preserved by proportionality.

Non-Euclidean geometry lacks similarity: for instance, doubling the lengths sides sharpens the angles of a hyperbolic triangle, and dulls those of a spherical one. So, even if we managed to get a pentagrammic "construction" of the Golden Ratio to work once ...

enter image description here

... we couldn't expect more: larger hyperbolic pentagrams have pointier points (contrariwise for spherical pentagrams), so that the ratio determined by how sides cut each other depends upon scale. Dependence is the antithesis of invariance.

That's not to say, however, that there's nothing to discuss. Indeed, I'm about to launch on a (ahem) lengthy digression ...

In non-Euclidean geometries "raw lengths", and/or ratios thereof, aren't a primary concern in establishing relations in triangles and other figures. Instead, lengths tend to be wrapped in (circular or hyperbolic) trig functions.

Consider, for instance, the Laws of Sines for a $\triangle ABC$ in Euclidean, Spherical, and Hyperbolic planes: $$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C} \qquad \frac{\sin a}{\sin A}=\frac{\sin b}{\sin B}=\frac{\sin c}{\sin C} \qquad \frac{\sinh a}{\sin A}=\frac{\sinh b}{\sin B}=\frac{\sinh c}{\sin C} \tag1$$ or the circumference of a circle with radius $r$: $$C_{euc} = 2\pi r \qquad C_{sph} = 2\pi \sin r \qquad C_{hyp} = 2\pi \sinh r \tag2$$ While it is true that the circumference-to-"raw"-radius ratio is not invariant, we can establish $\pi$ (and/or $\tau:=2\pi$) as an invariant "circle constant" by defining it in terms of the ratio of circumference and an appropriate transmutation of radius: $$2\pi = \frac{C_{euc}}{r} = \frac{C_{sph}}{\sin r} =\frac{C_{hyp}}{\sinh r} \tag3$$ Likewise for area: $$\pi = \frac{A_{euc}}{4\left(\tfrac12r\right)^2} = \frac{A_{sph}}{4\sin^2\tfrac12r} = \frac{A_{hyp}}{4\sinh^2\tfrac12r}\tag4$$

Such transmutation is precisely the mechanism that helps us establish the Golden Ratio as a scale-independent —and thus invariant— "pentagram constant" via the above construction, as we can write:

$$\phi = \frac{|KQ|_{euc}}{|PK|_{euc}} = \frac{\sin |KQ|_{sph}}{\sin |PK|_{sph}} = \frac{\sinh|KQ|_{hyp}}{\sinh|PK|_{hyp}} \tag{5}$$ where $|XY|_{geo}$ indicates distance in the corresponding geometry. (I'll omit the subscript when context is clear.)

Importantly, though, this kind of thing doesn't preserve every kind of ratio. In fact, many (most? all?) of the other occurrences of $\phi$ in the Euclidean pentagram don't transfer to non-Euclidean counterparts simply by swapping every length $\ell$ for $\sin \ell$ or $\sinh \ell$.

On the other hand, there are other interesting Euclidean occurrences of the Golden Ratio that do have direct non-Euclidean counterparts. For instance, here is a slight restatement of a delightful result from George Odom:

enter image description here

If a midpoint segment of an equilateral triangle is extended to meet the triangle's circumcircle at $P$ and $Q$, then each midpoint divides the extended segment in the square of the Golden Ratio. In particular, for $K$ in the figure, $$\frac{|QK|}{|PK|} = \phi^2 \tag{6}$$

(Odom actually says more-simply that, if $K'$ is the other midpoint, then $$\frac{|KK'|}{|PK|} = \phi \tag{7}$$ a relation equivalent to $(6)$ because $|QK|=|PK|+|KK'|$ and $\phi^2=\phi+1$.)

In hyperbolic geometry (spherical is analogous), the same construction "works", provided we apply the "appropriate transmutation" to the lengths:

enter image description here

$$\frac{\sinh|QK|}{\sinh|PK|} = \phi^2 \tag{8}$$

The best we can do for a counterpart to $(7)$ is $$\frac{\sinh|QK|-\sinh|PK|}{\sinh|PK|} = \phi \tag{9}$$ Notice that the numerator does not reduce to $\sinh|KK'|$; as I wrote, the transmutation trick doesn't work for all ratios. (This is why I had to restate Odom's result as a construction of $\phi^2$ instead of $\phi$.)

As a final example where the trick does work (and figures into two aspects of a result), there's this (re-discovered?) generalization of Odom's construction (blog link via trigonography.com):

enter image description here

$$\frac{|BK|}{|AK|} = 4\cos^2\theta \quad\iff\quad \frac{|QK|}{|PK|}=\phi^2 \tag{10}$$ which, eg, in the Hyperbolic plane becomes

enter image description here

$$\frac{\sinh|BK|}{\sinh|AK|} = 4\cos^2\theta \quad\iff\quad \frac{\sinh|QK|}{\sinh|PK|}=\phi^2 \tag{11}$$

I'm going to stop typing now. :)

  • $\begingroup$ Thank you very much for your answer. It looks like it was a lot of typing. Thank you for that work you did. I have not read it yet, only the first part. How does your answer relate to Pi? I am pretty certain the ratio of the circumference of a circle to its' diameter is dependent on the underlying geometry, and perhaps even on the location in the space. I think the arctic circle and the equator have different ratios to their longitudinal radii of the circles. I think the golden ratio is different than, say 1/2, though I can't say how, yet. I'll read and try to understand what you wrote. $\endgroup$ – Joe Cash Dec 12 '20 at 20:51
  • $\begingroup$ On further consideration, I think the difference between 1/2 and Pi or the golden ratio has something to do with relations. I can't explain why 1/2 is a different kind of relation than Pi is yet; but Pi does seem to vary depending on the underlying geometry. $\endgroup$ – Joe Cash Dec 12 '20 at 20:56
  • $\begingroup$ Sketch of a proof that the ratio of the circumference of a circle to the diameter of the circle is a different number in non-Euclidean geometries. See the next comment so I have enough room: $\endgroup$ – Joe Cash Dec 13 '20 at 20:21
  • $\begingroup$ Consider a ball with unit radius. The surface of the ball, a sphere, provides us with an elliptical (i.e. a non-Euclidean geometry.) Consider the equator of this ball as a circle with center at the north pole. The circumference is equal to two times the Euclidean number we call Pi ~= 3.14159.... The radius, however is equal to Pi/2, so the diameter is Pi. So, circumference / diameter = 2 * Pi / Pi = 2. So, in this geometry, the ratio of the circumference of a circle to the diameter of the circle is 2. $\endgroup$ – Joe Cash Dec 13 '20 at 20:26
  • $\begingroup$ Please remember, my motivation for asking the question was: What is invariant across both Euclidean and non-Euclidean geometries. Pi is not invariant. 1/2 seems to be invariant. All numbers may be invariant; so I'm not sure why I think the golden ratio is different than other numbers, yet. I'm not just doing what others have told me to believe in, I am exploring new (granted they are conjectural or even speculative) ideas here. This one, if it pays off, might be consequential. So, even with all the answers, the question remains: what is invariant across Euclidean and non-Euclidean geometries. $\endgroup$ – Joe Cash Dec 13 '20 at 20:33

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