#Braids #3StrandBraids

We are finally ready to convert the two #sinusoids from https://pixelfed.social/p/Splines/797893262102038801 into a single 3D curve that captures the essential geometry of a #braid strand.

First extrude the blue sinusoid into a surface that extends past the magenta sinusoid on both sides. Then draw a bounding box around the blue extrusion and trim the magenta sinusoid that falls outside the bounding box.

Discard the bounding box, and extrude the trimmed magenta sinusoid into a surface that extends past the blue extrusion on both sides.

Then split either surface with the other. It doesn't matter which surface is split and which is used as a cutting surface. The braid strand lies literally at the intersection of both surfaces.

I trimmed the magenta surface with the blue one and deleted the top portion to reveal the curve at the intersection — shown here in orange. In perspective view this curve continuously swerves from left to right and simultaneously from top to bottom as it progresses along the X axis.

This single curve has the characteristics of both sinusoids as seen in front and top views. In the side view, this looks like the #infinity symbol. So we have progressed from zero (with #helix), to plus (with #sinusoid), to infinity (with intersection of two #sinusoidal surfaces).

Once we have this curve, we can sweep a circle around it to make a round strand. We can change the radius of the circle to make thinner or thicker strands. We can slant the circles to give a "calligraphic" look to the strands. We can use ovals, rectangles, squares, stars, or any closed shape to give different surface properties to the strands — the possibilities are endless.

Once you have a closed #airtight strand with capped #planarHoles, make 2 more copies of the same strand. Shift the first copy by 1/3 the wavelength of the magenta sinusoid (48/3 = 16 units) and shift the second copy by 2/3 (48*2/3 = 32 units) while leaving the original one in its place.

#Braids #3StrandBraids

After creating the two #helix curves as described in https://pixelfed.social/p/Splines/797732962403957263, switch to the front view and #project the smaller blue helix on the vertical "wall" of the XZ plane. Hide the original helix. Then switch to the top view and project the larger magenta helix on the "ground" or XY plane and hide the original helix.

Now compare the figure in this post with that in the previous post. Both curves have now been #flattened from 3D helix to 2D #sinusoid. When viewed from the front (top-left portion of the diagram), the blue curve is still visible as a #sinusoidal waveform but the magenta appears as a straight line flattened on the ground.

When viewed from the top, the magenta curve is still visible as a sinusoid but the blue appears as a straight line clinging to the vertical wall. In the view from side (bottom-left portion of diagram), neither waveform is apparent, and both curves appear as perpendicular straight lines.

Only in the perspective view you can see both waveforms, but even here it is clear that they are both flat 2D curves oriented perpendicular to each other in 3D space.

Our goal is to convert these two flat sinusoids back into a single composite 3D curve that shows the smaller waveform in the front view and larger one in the top view.

In acoustics, a sinusoid represents a pure tone with a single frequency. The tone varies with frequency and its perceptibility varies with amplitude. Musicians and people familiar with acoustic physics will immediately recognize that the blue curve has twice the frequency (or pitch) of the magenta curve, while the magenta curve has twice the amplitude (loudness) of the blue curve.

We can divide the period or wavelength into phases. For the blue one, we divide the wavelength into 4 phases of 6 units each and shift the magenta curve left by that amount. Later, we will divide the magenta one into 3 phases — one for each strand, and shift each rightward by that.

#3StrandBraids

As I mentioned in https://pixelfed.social/p/Splines/797555432624206626, the geometry of braid strands is not at all obvious despite how familiar they look.

The geometry is defined in a series of steps starting with two coil-shaped #helix curves that we project on two perpendicular flat surfaces to create two flat #sinusoidal curves. Then, we transform the two flat sinusoids back into a composite 3D curve that acts as the rail for each strand. Then, we arrange three strands to form a braid. If you have never seen the geometry of a braid strand, you might be surprised at how it looks.

A braid strand is #periodic just like the #sinusoid curves it is based on. That means the pattern repeats at a fixed period and the motif can go on forever. For our purposes, a braid that is µ = 144 units long is sufficient, but to get that, we have to start with a helix that is longer than µ.

In the front view, start at the origin and draw a helix of length µ*4/3 = 192 units. This is shown in blue in the top-left portion of the figure. Directly underneath that is the view as seen from the top, which looks very similar to the one in the front view, but is distinct. Since a helix is like a round coil, it looks like a circle if viewed from a side as shown in bottom left. Only in the perspective view seen on the right is the coil shape readily apparent.

The blue helix has a fixed distance equal to 1/2 part (4 units) from the axis and it makes 8 full turns along the entire length of the axis, meaning it repeats with a period (or wavelength) of 192/8 = 24 units or 3 parts. This is crucial: For a 3 strand braid, the period must be a multiple of 3.

Duplicate the blue helix and double it in size while centered on the origin. Then, shift the larger helix by 1/4 of period (6 units) to the left. This larger helix is shown in magenta.

These helices are still in 3D. Make sure that the front, top, and right views look like what's shown here. Otherwise, the next step won't work.

#3StrandBraids

#Braids are the last of the #decorative elements on the #IonicScroll, but like #EggsAndDarts, they are not specific to the #IonicOrder.

Braids are a popular design motif that find wide currency in modern #hairstyle, #fashion, and fashion accessories like #belts and #bracelets.

Braids come in infinite varieties with varying number of strands, thickness of strands, roundness or flatness of strands, and how tightly or loosely they are wound together. Here, I focus on the 3-strand variant mentioned in #Vignola's book and previewed in https://pixelfed.social/p/Splines/792015485979791089. The image here is brightly colored to draw attention to the 3 strands.

The geometry of braid strands is not at all obvious despite how familiar they look. Also, a braid strand is the only feature in the entire iconic order whose geometry cannot be captured with straight lines and circular arcs. Instead, a strand geometry must be defined in a series of steps starting with a basic #sinusoidal curve.

A sinusoidal curve or #sinusoid is a wave form whose function belongs to a family of functions known as #transcendentalFunctions that also include #logarithmic and #exponential functions. I mentioned #logarithmicSpirals in https://pixelfed.social/p/Splines/792499765146596723, and in a future post I will show how to construct one and compare it with the #spiral used in our implementation of #IonicVolute.

They are called transcendental functions because they transcend the math of finite algebraic polynomials and go beyond geometry into trigonometry. Fortunately, we don't have to go there.

Few #CAD tools have a direct primitive for a sinusoid, but almost all have a primitive for a 3-dimensional round coil shape called a #helix which we can use to create the sinusoids we need for a braid strand. To create a sinusoid, all we need to do is #project a helix on a flat surface to convert it into a 2D waveform.