u/ArjenDijks

The Pentagon Ladder

A recursive geometry where nested squares, circles and pentagons scale with powers of the Golden Ratio (φ):
- Square = φ²ⁿ
- Circle = π φ²ⁿ / 4
- Pentagon = (5 φ²ⁿ/ 16) √(φ + 2)

• Base circles: c_d is the unit-diameter circle centered at (0.5, 0); c_r is the unit-radius circle centered at (0, 0).
• Seed point D: Moving D along c_d yields radius OD = cos(θ) (the cosine circle).
• Secant point S: Extending ray OD to the unit axis x = 1 gives point S, with radius OS = sec(θ) = 1/cos(θ) (the secant circle).
• Hyperbola: Drawing bounding square grids around both circles isolates rectangles with width cos(θ) and height sec(θ).

Because cos(θ) × sec(θ) = 1, the outer rectangle vertices H1, H2, H3, H4 directly plot the constant-area condition x × y = 1, tracing the rectangular hyperbola y = 1/x in real-time as D moves.

Vertices trace the Spiral Power Map unfolding and contracting. Areas and edge lengths scale recursively and exponentially.

I noticed that dividing a circle diameter into segments 2/3 and 1/3 produces a surprisingly rich configuration of right triangles and reciprocal roots.

From a single intersection point on the semicircle, one can trace a system of perpendiculars and transversals that naturally yields multiple √2- and √3-bearing segments through repeated applications of right-triangle geometry.

Rather than being computed algebraically, these roots emerge geometrically as distances, projections, and reciprocals.

I’m curious whether this generative approach, using a single division point to unfold a family of related roots, has appeared in pedagogical material, especially in generalizations of the form 1/n (for example, 4/5 and 1/5).

Has anyone seen this specific tracing method used for teaching radicals as geometric operators?

I’ve been exploring ways to visualize radical sequences similar to the Spiral of Theodorus.

By using a circle as a locus, you can map all square roots onto a single arc using perfectly even, linear steps along the x-axis.

Because the chord scales linearly with the diameter, we can "calculate" roots by just moving along a grid.

Interactive model: GeoGebra Animation

Questions: Has anyone seen this "chained chord" method used in historical drafting or nomography? Could this be extended to higher-order roots using other conic sections?

Full derivation/discussion on MSE: How to map square roots as a linear progression on a circle?