A Workshop on N-regular Polygon Torus using 4D Frame

Year: 2013 Authors: Park, Ho-Gul

Core claim

A 4D Frame can be used to construct n-regular polygon tori whose shapes converge toward a circle-torus as n increases.

Topics

torus geometry, polygonal frameworks, workshop construction, spatial beauty

Domains

geometry, topology, Pythagorean theorem, geometric sculpture, educational design, spatial craft

Methods

hands-on workshop, frame assembly, measurements, geometric calculation

Media

4D Frame, octagons, quadropod connectors, tubes

Paper text

The text below is the locally extracted OCR/Markdown version of the paper. Raw PDF files remain local and are not published here.

Proceedings of Bridges 2013: Mathematics, Music, Art, Architecture, Culture

A Workshop on N-regular Polygon Torus using 4D Frame

Park, Ho-Gul

4D Math and Science Creativity Institute

2F, Daegwang BD, 248-2, Gwangjang-Dong Gwangjin-Gu·Seoul·Korea

p3474@chol.com

Abstract

In this workshop we will show an n-regular polygon torus and its applications where $n=4$ , 8 using 4D Frame. We will bring the 4D Frame for participants to use. As $n$ approaches infinity, an n-regular polygon torus approaches to a circle-torus. Participants can make an 8-regular polygon torus (i.e. octagon-torus) by the Pythagoras theorem using 4D Frame’, consisting of 16 regular octagons. Therefore people will understand the simple mathematical structure and its spatial beauty in their work.

Figure 1: 8-regular polygon torus using 4D Frame

1. Introduction

A donut is an example of torus from daily life. (Figure 2) As you can see from Figure 3, we can find geometrically mathematical concept to inspire students.

Figure 2: a donut which is circle-torus

Figure 3: a donut structured

2. General Mathematical Definition of Torus

In mathematics a torus is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle. If the axis of revolution does not touch the circle, the surface has a ring shape and we call it a circle-torus.

Figure 4: a circle torus with a coplanar axis

Park

A torus can be defined parametrically as

where $\phi$ and $\theta$ are angles which make a full circle, starting at 0 and ending at $2\pi$ , so that their values start and end at the same point, R is the distance from the center of the tube to the center of the torus, r is the radius of the tube. Its surface area and interior volume are easily computed using the Pappus’ Centroid Theorem giving Area as $A=4{\pi }^{2}r\hslash$ and Volume as $V=2{\pi }^{2}R{r}^2$ .

3. N-regular Polygon Torus

Combining an n-regular polygon with a number of unit sets, we can make a donut shape (We call it an n-regular polygon torus). When $n=8$ , the process is suggested simply in Figure 5-a and 5-b. An 8-regular polygon torus consists of 16 regular octagons with quadropod connectors and tubes by 4D Frame (We also call it octagon-torus). (Figure 6)

Figure 5-a: 8-regular polygon $(C_1)$

Figure 5-b: 8-regular polygon torus $(C_2\sim C_6)$

• principal curve : C₁, C₂, C₃
• asymptotic curve : C₄
• geodesic curve : C₁, C₂, C₃, C₄

Let’s say principal curve $C_1$ and $C_2$ have 3 cm frames as you can see below. $U_1=3$ (cm), $U_2=3$ (cm)

Circumference of $C_1=24\mathrm{cm}$ … [Figure 6-a]

Circumference of $C_2=48\mathrm{cm}$ … [Figure 6-b]

Figure 6-a: $C_1$ using 4D Frame

Figure 6-b: $C_2$ using 4D Frame

Considering the length of the quadro pod(5cm), the circumference of $C_1$ and $C_2$ will be 28cm and 56cm.

$\therefore$ Diameter of $C_1:D_1=28÷3.14\approx 9(\mathrm{cm})$ $\therefore$ Diameter of $C_2$ : $D_2=56÷3.14\approx 18(\mathrm{cm})$ $\therefore$ Diameter of $C_3:D_3\approx 36(\mathrm{cm})$ … [Figure 7-a] $\therefore$ Circumference of $C_3:D_3\times 3.14\approx 113(\mathrm{cm})$ … [Figure 6-c]

A Workshop on N-regular Polygon Torus using 4D frame

Figure 7-a: Diameter of $C_1,C_2,C_3$

Figure 6-c: $C_3$ using 4D Frame

With the same method, we can get $C_3^{\prime }$ unit frame $U_3=(D_3\times 3.14-8)÷16\approx 6.5$ (cm) and $C_4^{\prime }$ unit frame $U_4=(D_4\times 3.14-8)÷16\approx 6.5$ (cm), $U_4\approx 4.8$ (cm) (Figure 7-b, 6-d).

Figure 7-b: Diameter of $C_4$

Figure 6-d: $C_4$ using 4D Frame

Let’s calculate the length of $C_5$ and $C_6$ by calculating $D_5$ and $D_6$ by the Pythagoras theorem. By the Pythagoras theorem, $\Delta OAB$ is a right-angled triangle.

With the same method,

Figure 8: Calculate $D_5$ from $C_1$

Figure 6-e: $C_5$ using 4D Frame

We can apply the torus by cutting half or quarter as in Figure 9-a, 9-b.

Park

Figure 9-a: a half 8-regular polygon torus (a half regular octagon torus)

Figure 9-b: combine two half regular polygon tori with 90 degrees

Similarly we can make a 4-regular polygon torus (a square torus) as shown in Figure 10 and Figure 11.

Figure 10: a square torus

Figure 11: application of a square torus

4. Conclusion and Perspectives

As $n$ approaches infinity, an $n$ -regular polygon torus approaches a circle-torus. Varying the sizes and the number of polygons we will get different and applicable open or closed torus structures as Figure 12.

Figure 12: a knot structure, application of torus, a knot structure in sphere, a Klein’s bottle

References

[1] http://en.Wikipedia.org/wiki/Torus. [2] Ho-Gul Park, The 3Rd Soil, 4D Frame 1st ed.(2006), 4D land Inc [3] http://www.4dframe.com [4] Andrew Pressely, Elementary Differential Geometry (2nd Ed), Springer [5] James R. Munkres Topology (2nd Ed), Prentice Hall