Acute Triangles Research Articles

Dissection of a Square into n Acute Isosceles Triangles

Dissection of a Square into n Acute Isosceles Triangles

Depth information in single triangles and arrays of triangles

When 6 obtuse triangles varying, in angular height from 75° to 10° were presented singly at 6 different slants, slant judgments were at chance level. When the view of an array of triangles was varied from 82° to 40° to 22° for the same 6 slants, the accuracy of slant judgments correspondingly varied. It was concluded that arrays of triangles carry information about their spatial orientation even though the individual elements of the array do not.

A Spherical Triangle Computer for Marine and Air Navigation

A simple analog computer, which solves the oblique and the right spherical triangle equation, is described based on the law of cosines. With three parameters of the oblique triangle as inputs, the fourth is computed with an accuracy better than ± 10 minutes of arc for most of the practical problems. The computer can be used to solve navigational problems such as the hour angle, course angle and great circle distance. Solutions for the right spherical triangle are obtained when one parameter is set to 90°.

A HYPOTHESIS FOR THE STRUCTURE OF RETAIL TRADE AREA

When the structure of activities of Retailing Business is comprehended as a regional phenomenon, it is usually approached by means of the analyses of its inner factors and outer fluctuating factors. In this article we shall attempt a dynamic approach to the latter.As one of the factors forming the Retail Trade Area, purchasing faculty was taken up and, to make it easy for us to compare, it was replaced by a purchasing rate and a purchasing faculty index. And on the bases of the data of the market survey which we made, according to the method of logical analyses from the standpoint of market geography, in Chigusadai Apartment Region in Nagoya City, in the groups of companies' houses in Yokkaichi City, and in the new city regions of Hanta City, Kasugai City and Ichinomiya City, we attempted the analyses of the purchasing rate and the purchasing faculty index, under the indexes such as income and occupations as social, economic conditions and the distance from a market as a geographical condition.As the result, the Retail Trade Area can be roughly divided into three zones, and its structure can be diagramed, with the distance or social economic index shown on the axis of ordinates and the purchasing rate on the axis of abscinassa. There can be seen an acute angled triangle, an epuilateral triangle, a pyramid, a cylinder or an upside-down pyramid. These shapes represent the structures of retail trade of the market, the department store and the specialty store one by one. And the commercial managing agency representing no model of trade area proves to be agency which has not settled its own trade area yet.In the diagram of the purchasing faculty index of final consumers classified by occupations, the salaried classed represent the highest structure. So we can realize that the most influential factor for the structure of consuming market is the salaried classes.The convenience goods and shopping goods represent the difference of social, economic, regional change in the distribution of purchasing rate. As for the convenience goods, the shopping chain relation of trade area between two cities competiting trade area can not be seen, but, as for the shopping goods, it can be seen.

The hyperfine structure of the V t-center

The structure of a new electron-deficient color center ( V-center) in LiF has been determined from its paramagnetic-resonance hyperfine spectrum. The electron deficiency is localized on three fluoride ions which form an obtuse isosceles triangle lying in a (001) plane with the height along a [110] direction. The orbital of the unpaired electron is primarily determined by the crystalline field and is of the π-type, in contrast to previously discussed V-centers. There is evidence that the electron deficiency is associated with an aggregate of three vacancies, so that the charge is compensated.

Mathematical Miscellanea

In Durell's Modern Geometry we find a surprisingly simple method of inscribing a square in a given triangle, based on the concept of homothetic1 figures. The method supplements two of the more conventional approaches to the problem. We are submitting three methods of inscribing a square in a triangle. No doubt there are more. Perhaps readers will provide additional approaches to the problem. We restrict our proofs to the acute triangle case.

The Infrared Spectrum of Chlorine Dioxide

The infrared spectrum of ClO2has been reinvestigated from 2–40μ. Several new bands have been discovered, among them the bending frequency ν2 at 445 cm−1. This band has the doublet character required if the molecular model is to be an obtuse triangle in accord with electron diffraction experiments. Three bands, ν1, ν2, and 2ν1, were resolved with prism and grating methods, and the spacing of the Q lines was found to be about 3.0 cm−1. An A″—B″ value of 1.413 cm−1 was obtained from the analysis of ν2, and this value also is in excellent agreement with the electron diffraction measurements. Moments of inertia computed from the r(Cl–O) = 1.49A and 2α (apex angle) = 118.5° obtained from a combination of electron diffraction and infrared data are IA=16.09×10−40 g cm2, IB=86.35×10−40 g cm2, and IC=102.43×10−40 g cm2. A band which cannot be ascribed to an impurity was found at 290 cm−1. The possibility that this band results from polymerization of ClO2 is suggested.

Mathematical Miscellanea

While browsing through the earliest French treatise on logarithms—Traicté des logarithmes (Paris, 1626) by Denis Henrion (c.1590-1640), Professor of Mathematics at Paris, my attention was caught by the method he adopted for the logarithmic solution of an oblique triangle when the three sides are given.

Triple-Dipole Interaction. I. Theory

Third-order perturbation theory is applied to the van der waals-type interaction between neutral atoms. An interaction between triplets of atoms results. The derivation of the third-order energy interaction W0′′′, is outlined, the procedure being somewhat similar to that used by London in his application of second-order perturbation theory to interatomic interaction. Since the perturbing potential is limited to the dipole-dipole term, the energy W0′′′ is called the triple-dipole interaction. The latter depends not only on the interatomic distances but also on the shape of the triangle formed by the three atoms; W0′′′ is positive for all acute and negative for most obtuse triangles.

Note on the Approximate Solution of an Oblique Triangle Without Tables

Note on the Approximate Solution of an Oblique Triangle Without Tables

Pitfalls in the Trigonometric Solution of an Oblique Triangle

Textbooks in plane trigonometry generally treat the solution of oblique triangles by the Law of Sines, Law of Cosines, etc. However, if one uses some of these laws without a complete understanding of their limitations, incorrect results are often obtained. The only case which most textbooks treat adequately in this respect is the solution of a triangle in which two sides and an angle opposite one of them are given, usually referred to as the “Ambiguous Case.” However, in other cases, such as the one in which two sides and the included angle are given, or the one in which the three sides are given, care must be taken if incorrect solutions are to be avoided.

Mammoth Tooth from Nokoro, South Sakhalin.

Lately the writer obtained a mammoth tooth found on a seabeach between Nokoro and Tirie in South Sakhalin.The degree of fossilization of the present specimen is fairly high, chocolate brown in colour with a strong lustre. In the crown view, the outerside is an acute triangle, and the side view is nearly a rectangle. It consists of seventeen ridges, excluding several ones which are inferred to have been lost on the occasion of digging. Of these the anterior ridges are almost straight in the sideview, while the posterior ones are more or less curved, some of them being S-shaped. The grinding surface is flat. The enamel layers are flexuous, but nearly straight; in this respect they differ from those of Manchurian Mammoth. This specimen is perhaps the third molar on the left mandible.Length of crown 205mm. Ridges 17Max. width of crown 80mm. Interspaces of ridges 5mmLength of grinding surface 115mm. Thickness of ridges 5mm±Height 120mm. Thickness of enamel layers 1.2mm±Height of crown 80mm . Frequency 10±The present molar is determined from these measurements to belong to Elephas primigenius (Blum). In many respects the sample has a close resemblance to that described by Mr. Sasa and thus it is considered to have been derived probably from the Rukutama Series of Upper Diluvial age.The present specimen will add one valuable material to the problem upon the palaeogeographical connection between Sakhalin and the Asiatic Continent in this age.

Investigations in the infra-red region of the spectrum. Part VIII.—The application of the grating spectrometer to certain bands in the spectra of triatomic molecules (Sulphur dioxide, and carbon disulphide)

In Part VI of the present series the spectroscopic similarity of the dioxides of sulphur and chlorine was demonstrated. The molecular structure of the two substances was left an open question, since the evidence then available gave contradictory results. The selection rules for the Raman spectrum demanded the obtuse-angled form, while, if complete resolution had been achieved in the infra-red bands, selection rules required the acute-angled modification. The present paper describes the reinvestigation of the bands in question using the grating spectrometer described in Part VII, and the results show that both sulphur and chlorine dioxides are obtuse angled triangles with a vertical angle of approximately 120°.

Form and Vibrational Frequencies of the NO2 Molecule

BAILEY and Cassie1 have recently presented data on the absorption of NO2 — N2O4 gas mixtures in the region from 2.5μ to 16μ, from which they con clude that the NO2 molecule is linear and symmetrical. As their conclusions and in some respects their data are in disagreement with results obtained recently in this Laboratory, and earlier results2, we wish to present briefly the facts that lead us to conclude that the atoms in the NO2 molecule are disposed as an obtuse angle triangle. Detailed reports will be published elsewhere.

A Solution of the Oblique Triangle Given Two Sides and the Included Angle

A Solution of the Oblique Triangle Given Two Sides and the Included Angle

A Solution of the Oblique Triangle Given Two Sides and the Included Angle

A Solution of the Oblique Triangle Given Two Sides and the Included Angle

Our Astronomical Column

NEW COMET.—A telegram from Kiel announces the observation of Comet Giacobini on September 4 last, at 8h. 44m. Nice mean time. Its position was then R.A. = 17h. 10m. 30s., Declination = -7° 29′. The given movement per day is 1˙8m. in R.A. and 0′˙25 in declination, so that the comet should be looked for soon after sunset, the above position being about 22° due south of α Herculis, making also a very obtuse isosceles triangle with ζ and v Ophiuchi. Another telegram gives particulars of a second observation of Comet Giacobini at Nice, on September 6, 8h. 26˙5m. Its position as observed was R.A. = 17h. 14m. 16s.; Declination = - 7° 49′. The comet is noted as very feeble.

VIII. On a new property of the tangents of the three angles of a plane triangle. By Mr. William Garrard, Quarter Master of Instruction at the Royal Naval Asylum at Greenwich. Communicated by the Astronomer Royal

Proposition I. In every acute angled plane triangle, the sum of the three tangents of the three angles multiplied by the square of the radius, is equal to the continued product of the tangents. Demonstration .—Let AH, HI, and IB be the arches to represent the given angles; and AG, HK, and BT be their & tangents, put r the radius, AG = a , and BT = b , Then r 2 / a and r 2 / b will be the tangents of HD and DI.