102.32 Panning for gold in the streams of a regular pentagon

2018 ◽  
Vol 102 (554) ◽  
pp. 332-335
Author(s):  
Tony Foster
Keyword(s):  
Regular Pentagon

Paper-folding and cutting activities to demonstrate five-fold symmetry

2018 ◽  
Vol 102 (555) ◽  
pp. 413-421
Author(s):  
King-Shun Leung
Keyword(s):  
Irrational Number ◽  
Simple Fraction ◽  
Regular Pentagon ◽  
Paper Folding ◽  
Pointed Star

We can obtain a two-fold symmetric figure by folding a square sheet of paper in the middle and then cutting along some curves drawn on the paper. By making two perpendicular folds through the centre of the paper and then cutting, we can obtain a four-fold symmetric figure. We can also get an eight-fold symmetric figure by making a fold bisecting an angle made by the two perpendicular folds before cutting. But it is not possible to obtain a three-fold, five-fold or six-fold symmetric figure in this way; we need to make more folds before cutting. Making a three-fold (respectively five-fold and six-fold) figure involves the division of the angle at the centre (360°) of a square sheet of a paper into six (respectively ten and twelve) equal parts. In other words, we need to construct the angles 60°, 36° and 30°. But these angles cannot be obtained by repeated bisections of 180° by simple folding as in the making of two-fold, four-fold and eight-fold figures. In [1], we see that each of the constructions of 60° and 30° applies the fact that sin 30° = ½ and takes only a few simple folding steps. The construction of 36° is more tedious (see, for example, [2] and [3]) as sin 36° is not a simple fraction but an irrational number. In this Article, we show how to make, by paper-folding and cutting a regular pentagon, a five-pointed star and create any five-fold figure as we want. The construction obtained by dividing the angle at the centre of a square paper into ten equal parts is called apentagon base. We gained much insight from [2] and [3] when developing the method for making the pentagon base to be presented below.

scholarly journals Golden Ratio, Silver Ratio and other Metallic Means ; Geometric Substantiation of all Metallic Ratios

2021 ◽  
Vol 20 ◽  
pp. 174-187
Author(s):  
Chetansing Rajput
Keyword(s):  
Golden Ratio ◽  
Geometric Features ◽  
Regular Pentagon

This paper introduces the concept of special right angled triangles those epitomize the different Metallic Ratios. These right triangles not only have the precise Metallic Means embedded in all their geometric features, but they also provide the most accurate geometric substantiation of all Metallic Means. These special right triangles manifest the corresponding Metallic Ratios more holistically than the regular pentagon, octagon or tridecagon, etc

See it, Change it, Reason it Out

1993 ◽  
Vol 40 (8) ◽  
pp. 434-436
Author(s):  
Jean M. Shaw
Keyword(s):  
Undergraduate Students ◽  
Regular Pentagon ◽  
University Museum

When we attended the Smithsonian Museum's traveling exhibition on kaleido-scopes at our University Museum, several undergraduate students made an interesting discovery. Pan of the exhibit featured a device with hinged mirrors and colored objects that could be arranged on a table; participants arranged the objects and explored the different effect that they could see reflected in the mirrors. When the students placed a colored strip between the ends of the mirrors and looked directly into the mirrors, which were positioned to create an angle with measure 90 degrees, they saw a square (fig. 1a). When they changed the angle of the mirror, they saw a regular pentagon or hexagon.

scholarly journals Calibrations for minimal networks in a covering space setting

2020 ◽  
Vol 26 ◽  
pp. 40 ◽  
Author(s):  
Marcello Carioni ◽  
Alessandra Pluda
Keyword(s):  
Minimization Problem ◽  
Covering Space ◽  
Regular Hexagon ◽  
Steiner Problem ◽  
Space Setting ◽  
Regular Pentagon ◽  
Minimal Networks

In this paper, we define a notion of calibration for an approach to the classical Steiner problem in a covering space setting and we give some explicit examples. Moreover, we introduce the notion of calibration in families: the idea is to divide the set of competitors in a suitable way, defining an appropriate (and weaker) notion of calibration. Then, calibrating the candidate minimizers in each family and comparing their perimeter, it is possible to find the minimizers of the minimization problem. Thanks to this procedure we prove the minimality of the Steiner configurations spanning the vertices of a regular hexagon and of a regular pentagon.

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