# Dissecting a brick into bars

@article{Feshchenko2008DissectingAB, title={Dissecting a brick into bars}, author={Ivan S. Feshchenko and Danylo V. Radchenko and Lev Radzivilovsky and Maksym Tantsiura}, journal={Geometriae Dedicata}, year={2008}, volume={145}, pages={159-168} }

Consider the set of all lengths of sides of an N-dimensional parallelepiped. If this set has no more than k elements, the parallelepiped will be called a bar (the definition of a bar depends on k). We prove that a parallelepiped can be dissected into a finite number of bars if and only if the lengths of its sides span a linear space of dimension at most k over $${{\mathbb Q}}$$ . This extends and generalizes a well-known theorem of Max Dehn about the splitting of rectangles into squares…

## 3 Citations

### Tiling a Rectangular Area Using a Set of Unique Squares

- Mathematics
- 2020

: - A set of natural numbers tiles the plane if a square-tiling of the plane exists using exactly one square of side length n for every n in the set. From [2] we know that N itself tiles the plane.…

### SQUARING AND NOT SQUARING ONE OR MORE PLANES

- Mathematics
- 2014

A set of natural numbers tiles the plane if a square-tiling of the plane exists using exactly one square of sidelength n for every n in the set. In [9] it is shown that N, the set of all natural…

### Possibilities and Impossibilities in Square-Tiling

- MathematicsInt. J. Comput. Geom. Appl.
- 2011

It is shown that any set growing faster than the Fibonacci numbers cannot tile the plane, and both tiling and non-tiling sets that can be partitioned into tiling sets, non-Tiling sets or a combination are found.

## References

SHOWING 1-10 OF 23 REFERENCES

### The Dissection of Rectangles Into Squares

- Mathematics
- 1940

We consider the problem of dividing a rectangle into a finite number of non-overlapping squares, no two of which are equal. A dissection of a rectangle R into a finite number n of non-overlapping…

### Tiling a square with silimar rectangles

- Mathematics
- 1994

In 1903 M. Dehn proved that a rectangle can be tiled (or partitioned) into finitely many squares if and only if the ratio of its base and height is rational. In this article we show that a square can…

### Tiling with Squares and Square-tileable Surfaces

- Mathematics

We introduce thèsquare tiling group' and use it to give necessary conditions on a planar polygon to be tileable with squares. We deene square tilings on Riemann surfaces, and compute the Euclidean…

### Tilings of the square with similar rectangels

- MathematicsDiscret. Comput. Geom.
- 1995

It is proved that the square can be decomposed into finitely many rectangles similar to R(u) if and only ifu is algebraic and each of its conjugates lies in the open half-plane Re(z)>0.

### Tiling a Square with Similar Rectangles

- Mathematics
- 2004

Theorem 1 If a finite number of rectangles, every one of which has at least one integer side, perfectly tile a big rectangle, then the big rectangle also has at least one integer side. Fourteen…

### Signed Tilings with Squares

- MathematicsJ. Comb. Theory, Ser. A
- 1999

LetTbe a bounded region in the Cartesian plane built from finitely many rectangles of the form [a1,,a2)×[b1,b2), with a1 a1 = b1, b2 a2 = a1 with respect to the plane.

### Filling boxes with bricks

- Materials Science
- 1969

• A submitted manuscript is the author's version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published…

### Delian Metamorphoses

- 2005

Nach einem Orakelspruch sollte die Pest in Griechenland dann zu Ende gehen, wenn der würfelförmige Altar im Apollonheiligtum auf Delos verdoppelt werde. Nach traditioneller Interpretation verlangt…