Rectangular Brick Research Articles

Bricklaying and the Hermite Normal Form

An integer lattice in n-dimensional space is the set of all integer linear combinations of n linearly independent vectors with integer entries. The translation group of the lattice consists of all the translations of n-dimensional space that take lattice points to lattice points. This group is isomorphic to the set of lattice points under vector addition. A fundamental brick will be a rectangular brick with one vertex at the origin and sides along each positive coordinate axis. A brickwork will consist of a tiling of n-space by n-dimensional bricks that are translations of a fundamental brick. The set of these translations form the translation group of the brickwork.

Experimental data for freezing and thawing of multi-dimensional objects

Testing the accuracy of freezing and thawing time prediction methods requires accurate experimental data. To complement existing data, 175 experimental measurements, 68 for thawing of rectangular bricks and 107 for both freezing and thawing of 12 different multi-dimensional irregular shapes, were made using Tylose, a food analogue, over a wide range of conditions. Twelve additional experiments were conducted using an actual food material, minched lean beef. Details of all the experimental conditions are reported.

Freezing Time Predictions for Brick and Cylindrical‐Shaped Foods

ABSTRACTA simplified model previously developed for freezing time calculations in plate freezers is extended to systems with two or three dimensional heat flow. The model combines Plank's equation with the unsteady heat transfer solutions for bodies with constant properties, through the addition of pre‐cooling, change of phase and tempering times. Average thermal properties, different for each period are used in order to take into account their change with the ice content along the freezing process. Freezing time predictions show a maximum difference of 10% with respect to freezing experiments performed with meat blocks shaped as cylinders or rectangular bricks. Processing times from 0.7–5 hr were compared with satisfactory agreement.

Algebraic theory of brick packing II

The general theory developed in an earlier paper [1] is applied to solve the problem of packing a region of n-dimensional space with rectangular bricks. When both positive and negative bricks are allowed, a complete solution is obtained for regions of arbitrary shape, while for packings with positive bricks only, a solution is given for sufficiently large rectangular boxes. Partial packings are also considered, and an asymptotic estimate is obtained for the wasted volume in the most efficient packing of a large box with a given set of bricks.

Two Simple Methods for Predicting Food Freezing Times with Time‐Variable Boundary Conditions

ABSTRACTTwo simple methods for predicting the freezing times of rectangular bricks, slabs, cylinders, and spheres in situations where boundary conditions change with time are proposed. These methods are based on numerical integration of a simple differential equation derived from a previously proposed modification to Plank's equation. The methods were tested against a three time‐level finite difference scheme for varying cooling medium temperatures and surface heat transfer coefficients. Agreement was generally good (difference < 10%) between the two methods and the corresponding finite difference solution.

PREDICTION OF FREEZING TIMES FOR FOODS IN RECTANGULAR PACKAGES

ABSTRACTA previous study of methods for predicting the freezing times of slabs, cylinders and spheres was extended to cover rectangular bricks of food materials. Experimental measurements of the freezing time were made over a wide range of conditions using a food analogue material, and these results compared to times calculated from various available prediction methods. The three‐dimensional version of a three time‐level finite difference scheme was found to be the most accurate numerical solution. Simple formulae are available for predicting the freezing time of rectangular bricks using the geometric factors derived by Plank. These factors are shown to be subject to error, and none of this group of methods gave accurate prediction of freezing time. A previously proposed group of formulae are extended to cover rectangular bricks. These formulae are simple to use and accurate. They are applicable to slabs (±5% with 95% confidence), and to cylinders and spheres (±7% with 95% confidence) as well as to bricks (±10% with 95% confidence). These extend the previously published formulae for slabs, cylinders and spheres without significant loss of accuracy.

Studies in Oriental Archaeology: I. Plano-Convex Bricks

DELOuGAZ has recently summarized our knowledge about pla-noconvex bricks and has advanced an hypothesis concerning their origin.' If I understand him correctly, he even offers two hypotheses which, however, are mutually exclusive. He says, p. 25: Their peculiar form did not come about through any architectural or technical need, but merely as a result of rather primitive craftsmanship ; and, p. 37: peculiar form resulted from the brickmaking technique. He has, indeed, shown beyond doubt that the rectangular lower part got its shape from the wooden frame into which the lump of clay was laid and that the upper bulge is the surplus of the rolled lump not taken off. But why was this surplus taken off in the preceding period and left in the planoconvex' period? The answer is obvious: the brickmaker wanted to retain this shape. Not the technique or craftsmanship is responsible for it, but the peculiar ideas of the architect who made some special use of the shape. I therefore think that Mr. Delougaz' second hypothesis is preferable, namely, that the plano-convex form imitates stone flakes for which such a shape is natural. This is the opinion also of other archaeologists, especially of Dr. Speiser.2 Dr. Speiser argues clearly and irrefutably as follows: Since planoconvex bricks replaced the much more practical former type of wholly rectangular bricks, and since the queer plano-convex shape