Alan J. CainFeb 2

The ‘Conics’ of Apollonius of Perga (c.260–c.190 BCE) became the standard text for ‘conic sections’ — the curves formed by the intersection of a plane and a cone, namely an ellipse, parabola, or hyperbola, depending on the angle of the plane relative to the slope of the cone (see attached image).

In the preface to the ‘Conics’, Apollonius wrote:

‘The third book contains many incredible theorems of use for the construction of solid loci and for limits of possibility of which the greatest part and the most beautiful [kallista κάλλιστα] are new.’

This quotation is triply important in the historiography of mathematical beauty: (1) it is the earliest extant description of a mathematical theorem as ‘beautiful’; (2) it is the earliest extant application of the term ‘beautiful’ to mathematics by a mathematician; and (3) it is the unique extant use of the term ‘beautiful’ to describe theorems by an ancient Greek mathematician.

(There is much discussion of the beauty of mathematics in ancient Greek thought, but it normally applies to the objects or concepts of mathematics.)

[Each day of February, I intend to post an interesting story/image/fact/anecdote related to the aesthetics of mathematics.]

1/3

#MathematicalBeauty #HistMath #Conic #ConicSection #geometry #aesthetics

Paysages MathématiquesOct 19
"If the Greeks had not cultivated conic sections, Kepler could not have superseded Ptolemy." – William Whewell (1794–1866)
#quote #mathematics #conic #maths #math
SplinesFeb 9, 2025
With the #secondaryCurves derived from #primaryCurves in https://pixelfed.social/p/Splines/794105734853818690, we are almost ready to sweep the #scroll surface. I say "almost" because there is at least one more refinement needed before we can use any of these curves.

Look at the front view of three sections of the scroll surface labeled A, B, and C, and you will see a qualitative difference among them. Surface A appears crude and surface C appears refined, while surface B lies somewhere in between. While B and C are both acceptable, A is not.

The difference is due to two factors — the nature of the curves themselves and the degree of precision used.

Surface A is built using the circular arc sections for #volute #spiral (original and scaled) as #railCurves and the secondary curve sections as #sweepingCurves. The nature of the two sets of curves is different. Straight lines are 1st-degree curves, #circular or #conic sections (including ellipse) are 2nd-degree curves, but the projected sweeping curves (secondary curves) are 3rd-degree #NURBS curves.

Sweeping 3rd-degree NURBS curves along 2nd-degree arcs does not produce a salubrious effect. So we #rebuild the arcs into a 3rd-degree curve using the #CAD tool. When we do that, we are able to control how close the rebuilt curve should be to the original arcs in terms of precision.

I rebuilt each arc in the spirals using 16 subsections, and the effect is visible in surface C.

Look at surface A again. The cross-section arcs appear unevenly spaced compared to those of the other surface sections. To fix that, I also rebuilt the projected NURBS curves (secondary curves) to obtain what I call #tertiaryCurves.

For the frontmost sections, I rebuilt the sweeping curves using 64 subsections, and for the rear sections, I rebuilt them with 8 subsections.

Experiment with what produces pleasing results, but remember that higher precision curves require more processing time as well as more storage space.
Splines (@Splines@pixelfed.social)

The #secondaryCurves derived in https://pixelfed.social/p/Splines/793641134563617634 with 4 #modulatingSpirals are sufficient for a rough draft when #3DPrinting, but sweeping the scroll surface using these curves still causes subtle wobbles. These wobbles generate undercuts that precludes #CNCMilling with 3-axis machines depending on orientation, and that requires 5-axis #CNC machines instead. To ameliorate that situation, I added 2 more interstitial frames labeled K and L, where k is 14 units in front of P, and L is 7 units behind Q. The size of K is 58.24 x 81.92 and that of L is 54.88 x 78.72. In other words, K is wider by 2.24 and taller by 1.92 compared to P and Q, while L is narrower by 1.12 and shorter by 1.28 compared to P and Q. K is offset from P in the front view by 0.64 at top, 1.28 at bottom, 1.44 at left, and 0.80 at right. L is concentric with Q in the front view with top and bottom insets of 0.64 and left and right inset of 0.56. How I derived these is too complicated to discuss within #Pixelfed character limits. Obviously, the scale factors for the spiral at K are 58.24/112 in X direction and 81.92/128 in Y direction. The scale factors for the spiral at L are 54.88/112 in X direction and 78.72/128 in Y direction. So, using these 6 modulating spirals, we again identify the tangent points with their respective frames and #project straight lines through these points on the scaffolding surface to get 6 higher-accuracy secondary curves. The diagram shows 6 blue #primaryCurves we extracted from #imageScans in https://pixelfed.social/p/Splines/793169876757012827 and https://pixelfed.social/p/Splines/793215298082967733 along with 6 new magenta secondary curves. The outlines we extracted from #Vignola’s antique images in 2-dimensions finally leap into 3-dimensions in a modern #CAD tool. The blue primary curves are no longer needed for this design, but don't discard them. They are beautifully proportioned and can be used in other designs.

Pixelfed
Paysages MathématiquesFeb 19, 2024
"If the Greeks had not cultivated conic sections, Kepler could not have superseded Ptolemy." – William Whewell (1794–1866)
#quote #mathematics #conic #maths #math
Fractal KittyOct 4, 2023

Day 4 - section:

when we see sections

of ourselves — our potential

manifests loci

https://codepen.io/fractalkitty/pen/wvRxORX

#mathober2023 #section #conic #hyperbola #p5js #procreate #codepen #math #mathart #loci #haiku #poetry

mathober day-4: Section

...

CodePen
Show threadDavid PierceApr 30, 2023

@bodhidave @EgyptianAphorist @mcmullin @clarablackink @histodons
Mathematics and #art are human activities, and in a #mathematics paper
(on #conic sections, the #conics, in particular, and described in the
linked blog post), I was pleased to have reason to quote what
Robert Pirsig said about teaching #writing. I commented,

> What #Pirsig wanted, as an English teacher, was for students to learn
> to write what *they* wanted. In the end, this would be what everybody
> else wanted, which was *quality.*

But there's a difference:

> In mathematics, an essential part of #quality is *truth,* or
> *correctness* if you prefer. We take this to be universal.

Not everybody may agree on what is good art, but I think they should
agree on what is correct mathematics.

https://polytropy.com/2020/08/05/an-exercise-in-analytic-geometry/

An Exercise in Analytic Geometry

Polytropy
SkyeFeb 13, 2023
Mensch und Berg - Man and mountain #Scotland #Balmaha #lochlomond #Conic

Walk: https://www.walkhighlands.co.uk/lochlomond/conic-hill.shtml
Conic Hill, Balmaha

Conic Hill is a sharp little summit rising above Balmaha. Right on the Highland Boundary Fault, this short hillwalk offers truly fantastic views over Loch Lomond and its many islands.

Walkhighlands
David PierceJan 29, 2023

Construction of points of a central #conic (here an #ellipse), made into a poster.

Details in my paper.

An earlier referee did wonder (before rejecting it): is it [really] art?

https://scholarship.claremont.edu/jhm/vol12/iss2/19/

Conic Diagrams

Textbooks may say that the so-called conic sections can be obtained from cones, but this is rarely proved. However, diagrams of the proof require no intuition for solids and can be read as flat. We construct the diagrams with ruler and compass and derive from them basic properties of conic sections as established by Apollonius of Perga, though again in a way that does not require a third dimension. The construction inevitably involves choices that give play to one’s aesthetic sense.

Scholarship @ Claremont
pablolarahNov 18, 2022

▪️▫️ Making Static Noise From a Weird CSS Gradient Bug
by Temani Afif @ChallengesCss
at @CSS

#backgroundclip #conic #gradients #gradients #mixblendmode #radialgradient #css #webdev

https://css-tricks.com/making-static-noise-from-a-weird-css-gradient-bug/