Ethnomathematics: Modelling the volume of solid of revolution at Buginese and Makassarese traditional foods

Ethnomathematics can empirically improve the cognitive abilities of students in elementary and secondary schools. However, in undergraduate study, there are still limited studies on integrating ethnomathematics in learning resources. This study aims to apply interpolation in modelling polynomial functions and integral volume on the shape of Buginese and Makassarese traditional foods. Furthermore, it can be used by students as relevant learning resources regarding interpolation and the concept of volume of solid of revolution (VOSR). This is a qualitative study using an ethnographic approach. The data were collected through observations to obtain general information, interviews with informants to find out food-making techniques, and documentation to obtain physical models of each type of food. Data Analysis Techniques consist of the domain analysis to obtain an overview of Buginese and Makassarese traditional foods and the taxonomic analysis to categorize mathematical concepts obtained from the modeling and simulation. The result of this research reveals that lammang is suitable with the slabs. It can be represented as constant functions that revolved around the x-axis or the y-axis. While paso, bolu cukke, and cantik manis as well as barongko batara, Putu, and cucuru can be outlined in linear functions rotating about the x-axis, y-axis, or others fixed-line. They meet the criteria of the disks method. However, they are described in the function of polynomials of n-degree. The use of washers can be described in the model of blundered and sarang semut with a hole in the middle caused by the intersection of two curves rotated about the x-axis or the y-axis. For shells, the model can be applied to determine the cover volume of the cover of pisang ijo flour and onde-onde. Thus, all types of traditional foods in this study can be appropriate objects for a learning resource in modelling the VOSR.

Accuracy of common stem volume formulae using terrestrial photogrammetric point clouds: a case study with savanna trees in Benin

Recent applications of digital photogrammetry in forestry have highlighted its utility as a viable mensuration technique. However, in tropical regions little research has been done on the accuracy of this approach for stem volume calculation. In this study, the performance of Structure from Motion photogrammetry for estimating individual tree stem volume in relation to traditional approaches was evaluated. We selected 30 trees from five savanna species growing at the periphery of the W National Park in northern Benin and measured their circumferences at different heights using traditional tape and clinometer. Stem volumes of sample trees were estimated from the measured circumferences using nine volumetric formulae for solids of revolution, including cylinder, cone, paraboloid, neiloid and their respective fustrums. Each tree was photographed and stem volume determined using a taper function derived from tri-dimensional stem models. This reference volume was compared with the results of formulaic estimations. Tree stem profiles were further decomposed into different portions, approximately corresponding to the stump, butt logs and logs, and the suitability of each solid of revolution was assessed for simulating the resulting shapes. Stem volumes calculated using the fustrums of paraboloid and neiloid formulae were the closest to reference volumes with a bias and root mean square error of 8.0% and 24.4%, respectively. Stems closely resembled fustrums of a paraboloid and a neiloid. Individual stem portions assumed different solids as follows: fustrums of paraboloid and neiloid were more prevalent from the stump to breast height, while a paraboloid closely matched stem shapes beyond this point. Therefore, a more accurate stem volumetric estimate was attained when stems were considered as a composite of at least three geometric solids.

ERRORS ANALYSIS OF STUDENTS IN SOLVING VOLUME OF THE SOLID OF REVOLUTION PROBLEM IN TERM OF CRITICAL THINKING ASPECTS

Students of mathematics education need to have a good understanding in solving integral calculus problems, but often have difficulty in understanding the subject material. The difficulties can be seen from the mistakes made by students when solving the problems. This article aims to analyze the errors of students in learning integral calculus especially in the subject of volume of the solid of revolution in terms of critical thinking aspects. This research is a qualitative descriptive study involving twenty-four of 6th-semester students of mathematics education at a private university in Yogyakarta. The research subjects were chosen purposively by considering: student errors in determining the volume of the solid of revolutions with integral techniques in terms of critical thinking aspect. The findings of this study indicate, student errors in determining the function of integrals, write down the rules of integral writing, misunderstanding in algebraic concepts, adding constants, write down limits and sigma notations, determine integration methods, determine boundaries integration, writing in integral sign, and error in determining the radius of the solid of revolution.

Examining Naturogenic Processes and Anthropogenic Influences on Tree Growth and Development via Stem Analysis: Data Processing and Computational Analytics

The objective of this study was to develop a stem analysis data processing and computational algorithm and associated software suite that was (1) applicable to temperate and boreal forest tree species, (2) mathematically consistent with excurrent tree stem geometric and allometric principles, (3) compatible with data structures obtained using proprietary and non-proprietary imaging systems, and (4) executable on Windows®-based operating systems. Computationally, the suite denoted SAP (Stem Analysis Program), deployed sectional-specific formulae that were in accord with the following geometric assumptions: (1) stump section was treated as a solid of revolution resembling a cylinder; (2) sections between the stump and the tip were treated as a solid of revolution resembling a frustum of a cone for sections with continuous annual increments, otherwise treated as a cone; and (3) tip section was treated as a solid of revolution resembling a cone. The algorithm also corrected for the slant-based sectional length measurements using Pythagorean Theorem and eliminated the requirement to predict age-specific apex height development through the use of a linear interpolation procedure. Based on input data structures consisting of annual ring-width xylem sequences measured from cross-sectional disk samples acquired at multiple positions along the tree’s main stem, the suite produces a broad array of output, inclusive of radial and longitudinal ring-width sequences, apical growth increments, annual and cumulative sectional and cumulative volume production patterns, and historically reconstructed stem taper profiles. In total, the SAP creates six output data files for each tree analyzed: (1) input data reference summary (e.g., geometric mean ring-widths and resultant radii for each cross-section); (2) radial growth patterns for the cross-section sampled at breast-height (e.g., absolute and relative diameter and basal area growth estimates); (3) sectional (vertical) profiles of volume growth patterns (e.g., absolute and relative growth estimates within each section (bolt)); (4) cumulative volume growth patterns for the entire tree; (5) historical taper profile estimates (e.g., heights and diameters by year); and (6) texturally-labeled compendium of all output files generated. Additionally, real-time graphical output was produced for the purposes of data assessment and verification during the radial sequence data acquisition stage (e.g., graphical presentation of annual ring-width sequences by radii and disk, for use in validating input data structures and increment measurements derived from the imaging system), and interpreting growth and development patterns (e.g., vertical growth layer and specific volume increment profiles by age or year). The utility of the SAP suite was exemplified by processing WindendroTM-based annual ring-width xylem sequences obtained from cross-sectional disks extracted from a jack pine (Pinus banksiana Lamb.) tree via percent-height destructive stem analysis, and subsequently elucidating growth and developmental patterns within the context of silviculture treatment effects (thinning). The SAP suite provides the conceptual and logistical foundation for the continued deployment of the stem analysis approach in a wide range of investigations, including those examining the effect of naturogenic processes and anthropogenic influences on tree growth and development.

REPRESENTATION OF SYNTHETIC SHADOWS ON 3D DIGITAL MODELS FOR THE VOLUMETRIC INTERPRETATION ON CONCAVE-CONVEX ARCHAEOLOGICAL ARTIFACTS

<p><strong>Abstract.</strong> The representation of archaeological artefacts aims for the graphic description of relevant information from the object, to allow for the proper interpretation of evidences from the past. Concavities and convexities are elements often difficult to represent through classical (analogical) representation techniques, especially when these geometrical characteristics are neither continue nor parameterizable. Digital techniques have advanced on the accurate reconstruction of 3D shapes, while attaching real colour to the geometry. However, the perception of concave/convex shapes from photorealistic true-orthoimages continues to be limited, especially when rich and homogeneous textures camouflage slight slope changes or volumetric deformations. In this paper we first critically review the current practice on the representation of solid-of-revolution artefacts with concave-convex predominance, and alternative photorealistic representations aiming at a better understanding of volume and colour. Given the limitation targeted, we then propose a workflow for the creation of true-orthometric maps enhanced by customized shadows. The work-flow integrates considerations on: (i) The orientation of archaeological artefact; (ii) the creation of accurate orthometric images based on Digital photogrammetry techniques; and (iii) the application of synthetic attached and cast shadows according the shape (information) to be represented. The workflow is demonstrated with a sample of plates retrieved from the Rua-das-Madres archaeological site, in Portugal.</p>

VOLUME AND TAPER EQUATIONS FOR COMMERCIAL STEMS OF Nothofagus obliqua AND N. alpina

ABSTRACT Timber volume of standing trees is essential information for management decisions. The increasing need to optimize the potential capacity of forests maintaining their conservation, requires the quantification of the different potential possible timber products. The aim was to adjust taper equations to determine volumes of different timber products for commercial stems of Nothofagus alpina and N. obliqua. Trees of both species were randomly selected in harvesting areas of Lanin National Park (Argentina). Trees were felled and cut into commercial logs, measuring diameter with bark at different heights up to the beginning of the crown, and for each tree the diameter at breast height and total height. Five taper equations were selected and non-linear regression processes were employed for the fittings. We obtained the volume through the integration of the stem profile equation and the rotation in the space thereof through solid of revolution. The Bennet and Swindel (1972) model was selected for both Nothofagus species, obtaining similar equation parameters and differences were observed at the top of the stems of larger trees. For this the use of an integrated model is not recommended. With the obtained equations it is possible to: (i) estimate volume at different heights and for different commercial diameters, and (ii) predict the height at which both species reach to a certain diameter. The model presented some statistical limitations (e.g. multicollinearity), however, the fitting of the equation and the easy understanding of the outputs support it as a useful tool in a broad range of forest applications.