Spiral Polygon Series Ralph Buchholz November 1985 - SMJ 31 Consider the two objects in Figure 1. On the left we see three equal squares made up of 12 unit edges while on the right we have four equal squares made up of 12 unit edges some of which are “shared”.

Figure 1: Squares with 12 edges This is in fact the basis of a well known problem in which one is asked to convert one configuration into the other by moving only three edges. Generalising this concept let us examine the following two problems :(a) What is the maximum number Smax (n) of unit edges required to construct n equal squares with no edges left over, and (b) What is the minimum number Smin (n) of unit edges required to construct n equal squares if sharing (of edges) is allowed. Clearly to maximise the number of edges we should just construct n disjoint squares each with 4 edges and thus Smax (n) = 4n. However to minimise the number of edges is not quite as easy. Consider the series of objects in Figure 2 which show a minimum configuration of edges to construct one to twelve squares. For the first square we can do no better than 4 unit edges. For the second and third squares we can share at most 1 edge each thus requiring 3 extra edges each. But for the fourth square we can share 2 edges thus requiring only two extra edges. Continuing in this manner we see that to obtain a minimising configuration for each successive square simply add a square in a spiral pattern. Note that some minimum configurations are not unique, for example three squares can also be constructed with 10 edges as in Figure 3. However, in general for n squares we must seek to minimise the perimeter (as the internal area will remain the same for different configurations of n squares). This can best be achieved by combining the squares to form a pattern as close as possible to a circle, hence the spiral algorithm (shown in Figure 4). Now to obtain the explicit formula for Smin (n) consider the case 1





 



 







 





Figure 2: Minimum Edges for 1 to 12 squares

Figure 3: Alternate minimum configurations for 3,7 and 8 squares n = m2 . Here the minimum configuration is always a complete large square made up of m × m unit squares. Referring to Figure 5 we see that the number of vertical unit edges is (m + 1) × m and similarly for the number of horizontal edges. Thus Smin (m2 ) = (m + 1)m + (m + 1)m = 2m2 + 2m. Now with this as a basis and assuming that the spiral algorithm does in fact provide the minimum configuration for every positive n one can prove that :√ Smin (n) = b2n + 2 nc where bxcdenotes the least integer greater than or equal to x. For example √ Smin (12) = b24 + 2 12c = b30.928c = 31. (c.f. Figure 2) Since both the equilateral triangle and regular hexagon can tile the infinite plane we can pose analagous questions to those for the square. Again we find that for both polygons a spiral pattern provides a minimum solution at each step (see Figures 6 and 7). As before we can extract the explicit formula for these two algorithms by first considering the results for each completed hexagon. For the equilateral 2

43 



   



21 

26 



 

 

37





7

 



    





5  

17  





2

3 





10  



 !#"%$&('*)+  ,.- /'+(' , 01+*"2/'3'54 6 - 7 , 01(*"/'3'58

  



13  

31

Figure 4: Square Spiral Algorithm triangle we have Emax (n) = 3n 3n Emin (n) = b + 2

r

3n c. 4

For the regular hexagon we have Hmax (n) = 6n Hmin (n) = b3n +

√ 12n − 3c.

Are these the only regular polygons for which we can pose this problem? What if we relax the “regular” restriction.

3



   !"

 $#%  &$'"! " )( Figure 5: Square Spiral Algorithm for perfect squares

                                      

         30

27

54

32

6

24

22

52

13

37

16

40

19

50

47

-/.0),#*)1 2"! #%"$'!#3&($ #*0)+)4")6#, 5 7 . #3 8!#9$ 0)+)1:

35

10

42

45

Figure 6: Triangular Spiral Algorithm   



  31



 

 

15

 28





17

 



 

34

6

20 



 

5 4



13  

22  



11 

9



3



 "!#%$'&)(*,+-!./ 0 12#$)+3% "!,#%$4&)"+-+35 076 "+/,+3 "!#$&)"+8+39 : 6 ;<% "!,#%$&)"+-+3=



8

25

Figure 7: Hexagonal Spiral Algorithm

4

Spiral Polygon Series

Spiral Polygon Series. Ralph Buchholz. November 1985 - SMJ 31. Consider the two objects in Figure 1. On the left we see three equal squares made up of 12 ...

144KB Sizes 2 Downloads 174 Views

Recommend Documents

5.02 Polygon practice
1. interior angle sum: S = 180°(n–2). 2. exterior angle sum of convex polygon: 360°. 3. each interior angle of a regular polygon: Each interior = 180°(n − 2) n. 4. each exterior angle of a regular polygon: Each exterior = 360° n. 1. Find the

pdf-61\aaa-spiral-barcelona-aaa-spiral-guides ...
Page 1 of 8. AAA SPIRAL BARCELONA (AAA SPIRAL. GUIDES: BARCELONA) BY ANDREW. BENSON, TERESA FISHER, CLARISSA. HYMAN. DOWNLOAD ...

SPIRAL FINAL.pdf
and one play by an American dramatist.). ... the effects of techniques unique to each medium (e.g., lighting, sound, color, ... visual and multimedia elements contribute to the meaning, tone, or beauty of a text (e.g., graphic .... SPIRAL FINAL.pdf.

Polygon HW#1.pdf
Sign in. Loading… Whoops! There was a problem loading more pages. Retrying... Whoops! There was a problem previewing this document. Retrying.

pdf-1865\polygon-mesh-processing.pdf
Sign in. Loading… Whoops! There was a problem loading more pages. Retrying... Whoops! There was a problem previewing this document. Retrying.

Polygon Pictures and Partners Present ZIGURUHAZERU Toy Line ...
Apr 5, 2013 - Connect pieces from all three characters to build the mega puzzle robot Hazeru Seioh. ... main business focus is on providing digital content creation services. ... For more information, visit our website at http://www.ppi.co.jp.

Spiral Review KEY.pdf
fnz. fn< 5. f\{. a. Algebra Find tlre lerrgtlt nf tlte rrt. 18. r;*3. n +3 t ?t,r' *€, : r+L. :, r. i i-e *n-1. ffii l1l{:ffs"r ' I. \*.* I. ALGEBRA I?Sft.I is n rectangle. 20. If nr r s[,iT : l"r * B ...

(Hcpcs Level II Expert (Spiral))
settings with the Optum360. HCPCS Level II Expert. Use this comprehensive reference for the HCPCS code set that focuses on management of reimbursement. ... submitted under the system to durable medical payers. In- depth illustrations. Enhance your co

The spiral galaxy blowup of SG - MOBILPASAR.COM
Aug 10, 2016 - Outline. 1. Motivation and background. 2. The Laplacian on SG∞. 3. Heat kernel on SG∞. 4. Schrödinger operators on SG∞. Andrew Hahm, Jeffrey Kuan. The spiral galaxy blowup of SG. August 10, 2016. 2 / 56 ...

Background information IPBES stakeholder engagement.pdf - Spiral
Intergovernmental Platform on Biodiversity and Ecosystem Services (IPBES). The engagement of ... peoples and local communities and the private sector.

Spiral of Violence diagram.pdf
Page 1 of 1. www.ChedMyers.org for Pacific Presbytery, September 2014. Violence # 1 Structural, Generative. Violence # 2 Reactive: Introjected or Projected.

Polygon Pictures and Partners Present ZIGURUHAZERU Toy Line ...
Apr 5, 2013 - ... Pictures Inc. Address: 1F Azabu Green Terrace, 3-20-1 Minami-Azabu, Minato-ku, Tokyo 106-0047. Page 3. Email: [email protected].

A Simple, Fast, and Effective Polygon Reduction ...
method by which your engine can quickly reduce polygon counts at ..... search all neighboring edges for “least cost” edge ... ferent pieces to optimize for human.

Polygon-Connected Autotransformer Based 28- Pulse ...
Abstract-- This paper presents the design and analysis of a. Polygon-Connected autotransformer based 28-pulse ac-dc converter which supplies direct torque ...

A Simple, Fast, and Effective Polygon Reduction Algorithm - Stan Melax
Special effects in your game modify the geometry of objects, bumping up your polygon count and requiring a method by which your engine can quickly reduce polygon counts at run time. G A M E D E V E L O P E R. NOVEMBER 1998 http://www.gdmag.com. 44. R

spiral dynamics integral pdf
Page 1 of 1. File: Spiral dynamics integral pdf. Download now. Click here if your download doesn't start automatically. Page 1 of 1. spiral dynamics integral pdf.

Spiral of Silence - A First Look
heat during the final two months of the campaign. But according to Elisabeth ... media in the process we first must understand people's extraordinary sensitivity ... equipped with antennae that quiver to every shift in the social breeze. ..... who fo

MODELING OF SPIRAL INDUCTORS AND ... - Semantic Scholar
50. 6.2 Inductor. 51. 6.2.1 Entering Substrate and Layer Technology Data. 52 ... Smith chart illustration the effect the of ground shield. 75 with the outer circle ...