Same thing
An even thickness or an area of constant width is clearly a figure that is equally thick or has the same width everywhere. The edge of such a figure is called a curve of constant width or orbiform ("circular").
definition
The width of a curve is defined as the distance between two parallel straight lines that touch the curve on opposite sides. These straight lines are called supporting straight lines . Curves of constant width are those curves in which the same value always results for the distance between these straight lines, regardless of the point on the figure at which the straight lines attack.
Examples and characteristics
The simplest example for a Gleichdick is a circle , this is the trivial case. The simplest nontrivial of all equidistance is the Reuleaux triangle . It is the same thickness with the smallest area , whereas the circle is the one with the largest, in between there are infinitely many others. So you can save material with the same thickness: a circle with the same diameter has a larger area, a cylindrical roller has more volume.
A constant thickness does not have to be made of circular arcs or be somehow symmetrical. All of the same thicknesses have in common the convex shape.
According to Barbier's theorem, the following applies to the circumference of any constant width :
That is, the circumference of a uniform thickness is equal to the circumference of a circle with the same diameter.
Well-known examples of a uniform thickness are the British 20 and 50 pence coins from the issue date 1982 and 1969 respectively (until current). Its heptagonal shape with rounded sides means that material is saved compared to the circle with the same diameter. The coin diameter can (apart from an error due to the rounded corners) be determined between parallel jaws or a coin slot in any direction of the coin. The corners can be felt when turning between the fingers. If the coins roll fast enough, they can lift or jump off a hard track.
A drill with the cross-section of a Reuleaux triangle can be used to drill "square" holes. This drill, which produces almost square holes, was invented by the British engineer Harry James Watt in 1914 (US Patent 1241175 and following).
The three-dimensional generalization
A spatial uniform thickness is a convex body of constant width: a body without indentations that always touches all six side surfaces in any position within a suitable cube: in whatever orientation such a body is clamped between two parallel plates, the two plates are always exactly the same far apart from each other.
A simple nontrivial spatial constant is the solid of revolution, which is created by rotating a Reuleaux triangle around one of its axes of symmetry. But also all other Reuleaux polygons rotated around an axis of symmetry are bodies of constant width. This means that there are infinitely many different spatial uniform thicknesses of the same constant width. The ratio between inscribed circle and circumference is also called trilobularity after the trilobular .
Contrary to the intuitive assumption that the Reuleaux tetrahedron is also of constant width, this body is not a uniform thickness. However, on this basis, it is possible to construct spatial uniform thicknesses that are not bodies of revolution, the two Meissner bodies .
Occurrence of equal thicknesses in production
When (circumferential) rolling of cylindrical or cylinder-like workpieces such as screws or plug-in contacts , tolerances make it more the rule than the exception that instead of cylinders, uniform thickness is created. This is often not critical, but problems arise when a form fit (e.g. for tightness) is required.
By measuring between two parallel surfaces, for example with a caliper or a jaw-shaped gauge , whose parallel jaws each determine the diameter measured only at two points, no difference can be determined due to the definition of the constant thickness. However, if a measuring instrument with three sensing points arranged approximately in the form of an equilateral triangle is brought up to a Reuleaux triangle, different dimensions can be detected depending on the rotational position of the workpiece. Such measuring devices can be implemented with a caliper, one jaw of which is placed an M-shaped fitting piece (with 120 ° V or rectangular groove) or two circular disks at a constant, suitable distance from one another in the measuring direction.
In contrast to the above-mentioned drill , which was specially invented for drilling square holes, a similar effect can occur unintentionally with any drill if lateral movement of the drill tip cannot be completely prevented. Drills, which are easier to evade because of their length and only have two short cutting edges, tend to jerk out a Reuleaux triangle instead of a circle . Drill bit with three cutting edges tend to a rounded four drill out eck (see picture), etc. In order to prevent non-circular boreholes, one can first pre-drill the hole with an approximately quarter to one-third of the final diameter. The pilot hole helps to guide the next drill centrally. With twist drills , the surfaces lying on a cylinder behind the spiral cutting edges guide the drill from a certain penetration depth and thus help to prevent out-of-round drilling.
Screws in sheet metal or plastic self-locking should have property to twisting, are sometimes with along the circumference in three places slightly bumpy protruding cutting iron produced, for which the contour of a Reuleaux triangle is suitable. It is typical that these humps lie as a triple helix on the inside of the circumscribed cylinder. The air in these often self-tapping screws opposite the cylindrical thread that is formed offers space for taking up chips . In particular, plastic is advantageously tensioned resiliently by expanding at the three triangular points.
See also
literature
- Günter Aumann : Circular Geometry: An Elementary Introduction . Springer, 2015, ISBN 978-3-662-45306-3 , pp. 207-230
- Hans Rademacher , Otto Toeplitz : Of numbers and figures: Samples of mathematical thinking for lovers of mathematics . Springer, 2nd edition 1933, pp. 137-150
- Christian Blatter : Over curves of constant width . In: Elements of Mathematics , Volume 36, Issue 5, 1981, pp. 105–114
- Karl Strubecker : Curve theory of plane and space . Walter de Gruyter (Göschen Collection, Volume 1113), 1955, pp. 51–55
- Lucas Geitel: Gleichdicks - figures of constant width . In: Alexander Blinne (Hrsg.), Matthias Müller (Hrsg.), Konrad Schöbel (Hrsg.): What would mathematics be without the root ?: The most beautiful articles from 50 years of the journal Die Wurzel . Springer, 2017, ISBN 9783658147594 , pp. 263–268
- Julian Havil : Curves for the Mathematically Curious: An Anthology of the Unpredictable, Historical, Beautiful, and Romantic . Princeton University Press, 2019, ISBN 9780691197784 , pp. 104-125
Web links
- Animations and explanations on the flat equal-thickness
- Further remarks on the three-dimensional generalization, in particular films of the two Meissner bodies
- Immediately on mathematische-basteleien.de
- Same thing
- SELF-PARALLEL CURVE, CURVE OF CONSTANT WIDTH on mathcurve.com
Individual evidence
- ↑ Thomas Andreas Peter: Characterization of osteosynthesis plates (PDF), dissertation at the ETH Zurich as a doctor of technical sciences, 2001, p. 139 f.