scholarly journals Hidden in Plane Sight: The Extraordinary Vision of Ernest Irving Freese

2019 ◽  
Vol 86 (11) ◽  
Author(s):  
Greg N. Frederickson
Keyword(s):  
Equilateral Triangle ◽  
Regular Hexagon

Abstract We assess selected geometric dissections associated with, or inspired by, Ernest Irving Freese’s 1957 manuscript and identify noteworthy features that lay beneath their surface. These include a dissection of a regular dodecagon to a regular hexagon, a hingeable dissection of a Greek Cross to an equilateral triangle, a hingeable dissection of an equilateral triangle to a regular hexagon, a hingeable dissection of ten regular pentagons to a decagonal ring, a translational dissection of two regular decagons to one, a translational dissection of a dodecagram and a co-dodecagram to a regular dodecagon, a translational dissection of ten regular pentagons to two 5-pointed pseudo-stars and a regular decagon, a translational dissection of ten regular heptagons to four 7-pointed pseudo-stars and a regular 14-gon, a translational dissection of three regular octagons to a larger octagon, a hingeable dissection of regular pentagons for (sin π/5)2 + (cos π/5)2 = 1, a translational dissection of squares of areas 1, 3, 5, 7, 9, and 11 to a large square, and a translational dissection of squares for 82 + 92 + 122 = 172. Unsubstantiated claims by Freese are also evaluated.

Semiregular Tessellations with Pattern Blocks

2017 ◽  
Vol 111 (2) ◽  
pp. 90-94
Author(s):  
Debananda Chakraborty ◽  
Gunhan Caglayan
Keyword(s):  
New Jersey ◽  
Equilateral Triangle ◽  
Number Sense ◽  
Wall Painting ◽  
Regular Hexagon ◽  
Jersey City ◽  
Instructional Tools ◽  
Isosceles Trapezoid ◽  
Measurement Algebra ◽  
Geometry Measurement

Pattern blocks are multifunctional instructional tools with a variety of applications in various strands of mathematics (number sense, geometry, measurement, algebra, probability). The six pattern blocks are an equilateral triangle (green), a blue rhombus, an isosceles trapezoid (red), a regular hexagon (yellow), a square (orange), and a white rhombus. The sides of all pattern blocks are congruent, considered to be 1 unit in length for this article. Photograph 1 depicts a wall painting with squares and rhombuses found in Jersey City, New Jersey.

Generalized Second Moment of Areas of Regular Polygons for Ludwick Type Material and its Application to Cantilever Column Buckling

2019 ◽  
Vol 19 (02) ◽  
pp. 1950010
Author(s):  
Joon Kyu Lee ◽  
Byoung Koo Lee
Keyword(s):  
Cross Sections ◽  
Equilateral Triangle ◽  
Regular Polygon ◽  
Regular Hexagon ◽  
Type Material ◽  
Regular Polygons ◽  
Column Buckling ◽  
Second Moment ◽  
Regular Pentagon

This study deals with the generalized second moment of area (GSMA) of regular polygon cross-sections for the Ludwick type material and its application to cantilever column buckling. In the literature, the GSMA for the Ludwick type material has only been considered for rectangular, elliptical and superellipsoidal cross-sections. This study calculates the GSMAs of regular polygon cross-sections other than those mentioned above. The GSMAs calculated by varying the mechanical constant of the Ludwick type material for the equilateral triangle, square, regular pentagon, regular hexagon and circular cross-sections are reported in tables and figures. The GSMAs obtained from this study are applied to cantilever column buckling, with results shown in tables and figures.

scholarly journals Regular-hexagon-equilateral-triangle area grouping–based broadcast protocol for safety message in urban vehicular ad hoc networks

2017 ◽  
Vol 13 (1) ◽  
pp. 155014771668382
Author(s):  
Xiufeng Wang ◽  
Mingcheng Qu ◽  
Gang Cui ◽  
Moshe Yan ◽  
Nwe Nwe Htay Win ◽  
...  
Keyword(s):  
Ad Hoc Networks ◽  
Waiting Time ◽  
Equilateral Triangle ◽  
Vehicular Ad Hoc Networks ◽  
Ad Hoc ◽  
Relay Node ◽  
Regular Hexagon ◽  
Grouping Method ◽  
Hoc Networks ◽  
Broadcast Protocol

Road-based directional broadcast protocols are proposed in the literatures to offset efficiency of message dissemination of traditional broadcast protocol in urban vehicular ad hoc networks. However, these protocols cannot provide enough reliability and efficiency for vehicles’ misclassification at intersection or on straight road. Therefore, we present regular-hexagon-equilateral-triangle area grouping–based broadcast protocol for urban vehicular ad hoc networks. The area covered by relay node is averagely divided using regular-hexagon-equilateral-triangle, and vehicles are grouped according to the area that they reside. The algorithm for constructing regular-hexagon-equilateral-triangle is proposed. We adopt same vehicles’ grouping method at intersection and on straight road, and no neighbor list is maintained to identify road intersection. We design waiting time formula to calculate time for node forwarding message. It is ruled that node with waiting time dropping to zero first is defined as the relay node, and this relay node transmits message. So there is only one relay node which forwards message in each group. It also rules in regular-hexagon-equilateral-triangle area grouping–based broadcast protocol that each relay node forwards the same message only once, therefore, it limits redundant message retransmission. Using the vehicles’ grouping method and selection strategy of relay node as mentioned above, the proposed protocol enables message to be transmitted in different directions along different roads at the same time. Simulation indicates that our protocol has a better performance

Finding the Area of a Circle: Use a Cake Pan and Leave Out the Pi

1986 ◽  
Vol 33 (9) ◽  
pp. 12-18
Author(s):  
Walter Szetela ◽  
Douglas T. Owens
Keyword(s):  
Preservice Teachers ◽  
Equilateral Triangle ◽  
Regular Hexagon ◽  
Common Misconceptions

The fact that many students have misconceptions and misunde rstandings of the concept of area has been clearly documented (e.g., Carpenter et al. 1980; Jamski 1978). For example, one of the most common misconceptions held by students in grades seven and eight and even preservice teachers is that polygons with the same perimeter have the same area. Szetela (1980) found that fifty-four of ninety-four (57 percent) stude nts in grades seven and eight, when presented with diagrams of a regular hexagon and an equilateral triangle known to have the same perimeter, stated that the areas of the two regions were the same despite the fact that the hexagon contained 50 percent more area than the triangle. Even more astonishing are the results of a similar experiment by Woodward and Byrd (1983). They found that 157 of 258 (61 percent) students in grade eight stated that the areas of five different rectangles with the same perimeters were the same even though one of the rectangles was almost four times as large as another! If students have difficulty with the concept of area of polygonal figures, they can be expected to have even more difficulty with areas of irregular figures, “blobs,” and circular regions.

scholarly journals GERBERT OF AURILLAC: ASTRONOMY AND GEOMETRY IN TENTH CENTURY EUROPE

2013 ◽  
Vol 23 ◽  
pp. 467-471 ◽  
Author(s):  
COSTANTINO SIGISMONDI
Keyword(s):  
Rational Approximation ◽  
Equilateral Triangle ◽  
Tenth Century ◽  
The Sun ◽  
Astronomical Observations ◽  
The North ◽  
Celestial Pole ◽  
Gerbert Of Aurillac

Gerbert of Aurillac was the most prominent personality of the tenth century: astronomer, organ builder and music theoretician, mathematician, philosopher, and finally pope with the name of Silvester II (999–1003). Gerbert introduced firstly the arabic numbers in Europe, invented an abacus for speeding the calculations and found a rational approximation for the equilateral triangle area, in the letter to Adelbold here discussed. Gerbert described a semi-sphere to Constantine of Fleury with built-in sighting tubes, used for astronomical observations. The procedure to identify the star nearest to the North celestial pole is very accurate and still in use in the XII century, when Computatrix was the name of Polaris. For didactical purposes the Polaris would have been precise enough and much less time consuming, but here Gerbert was clearly aligning a precise equatorial mount for a fixed instrument for accurate daytime observations. Through the sighting tubes it was possible to detect equinoxes and solstices by observing the Sun in the corresponding days. The horalogium of Magdeburg was probably a big and fixed-mount nocturlabe, always pointing the star near the celestial pole.

The Optimum Quality Index for the Stability of In-Parallel Planar Platform Devices

1999 ◽  
Vol 121 (1) ◽  
pp. 15-20 ◽  
Author(s):  
J. Lee ◽  
J. Duffy ◽  
M. Keler
Keyword(s):  
Optimal Design ◽  
Equilateral Triangle ◽  
Quality Index ◽  
Optimal Shape ◽  
Geometrical Meaning ◽  
Moving Platforms ◽  
Maximum Value ◽  
Dimensionless Ratio ◽  
Parallel Platform ◽  
The Stability

The paper investigates primarily the geometrical meaning of the determinant of the Jacobian (det j) of the three connector lines of a planar in-parallel platform device using reciprocity. A remarkably simple result is deduced: The maximum value of det j namely, det jm is simply one-half of the sum of the lengths of the sides of the moving triangular platform. Further, this result is shown to be independent of the location of the fixed pivots in the base. A dimensionless ratio λ = |det j|/det jm is defined as the quality index (0 ≤ λ ≤ 1) and it is proposed here to use it to measure “closeness” to a singularity. An example which determines the optimal design by comparing different shaped moving platforms having the same det jm is given and demonstrates that the optimal shape is in fact an equilateral triangle

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