Topology-disturbing objects: a new class of 3D optical illusion, Hacker News

31 August 2017

(Figure 3.Topology-disturbing triplet of cylinders.

Figure 3 .Topology-disturbing triplet of cylinders.

Figure 4shows another type of a topology-disturbing object. In the direct view, the two cylinders are concentric (nested). However, in the mirror, the cylinders appear to have changed shape and to intersect each other.

(Figure 4.Topology-disturbing nesting cylinders.

(Figure 5) shows another object, which consists of a cylinder and a plane. In the direct view, the plane passes through the cylinder, while in the mirror, the plane and the cylinder are separated.

Figure 5.Topology-disturbing cylinder and a plane .

The above examples are typical topology-disturbing objects. Each object consists of two or more cylinders (Figure 5includes a plane). In one view, they are disconnected, while in the other view, they intersect. Disconnection and intersection are topologically invariant properties, but in this case, they are not preserved in the mirror image, and thus we call them topology-disturbing objects.

3. Principle of ambiguous cylinders

Topology-disturbing objects can be constructed by applying a variation of the method used to design ambiguous cylinders. Hence, in preparation, we will briefly review the principles of ambiguous cylinders [(K.) Sugihara,Ambiguous cylinders: A new class of impossible objects, CADDM. 25 (3) (2015), pp.19–25.[Google Scholar]]. An ambiguous cylinder is a cylindrical object that appears to have different structures when viewed from two special directions. They can be constructed in the following way:

As shown in (Figure 6) , we fix anxyzCartesian coordinate system so that thexy−plane is horizontal, and the positivezdirection orients upward. LetV(1)=(0, cos θ, −sin θ) and(V)(2)=(0, −cos θ, −sin θ) be two viewing directions; they are parallel to theyz−plane, and they are directed downward at the same angle θ, but in opposite directions.

Figure 6.Two viewing directions parallel to the (yz) – plane.

Next, as shown inFigure 7, on thexy−plane, we fix two curves (a) (x) andb(x) forx(0) ≤ (x) ≤x(1) . Note that the initial pointsa( (x) (0) ) andb(x₀) have the samex−coordinate, and similarly, the end points (a) (x(1) ) and (b) ****************************** (x(1) ) have the same (x) −coordinate. Moreover, these two curves arex– monotone. For eachx, (x) (0) ≤x≤x(1) , we consider the line passing through (a) ( (x) ) and parallel to(v)(1) , and the line passing through (B) (x) and parallel to(v)(2) . These two lines are included in the same plane parallel to theyz−plane, and hence they have a point of intersection. Let this point be denoted byc(x). Then,c( (x) ),x₀≤ (x) ****************************** (≤) ***************************** (x) (1) , forms a space curve. The curvec( (x) ) coincides with (a) ****************************** (x) when it is seen in the direction(v)(1) , and it coincides with (b) (x) when it is seen in the directionv(2) .

Figure 7.Space curve whose appearances coincide with two given plane curves.

Finally, we choose a vertical line segment (L) and move it in such a way that it remains vertical, and the upper terminal point traces along the curvec(x), (x) (0) ≤ (x) ≤x(1) . LetSbe the surface swept by (L) .Sis a surface with vertical rulers, and the vertical length ofSis the same as the length of (L) . Therefore, when viewed, it appears to be a cylindrical surface with a constant height. In other words, we are likely to perceive it as a cylindrical surface whose upper and lower edges are obtained by cutting the surface with a plane perpendicular to the axis of the cylinder. This may be the result of the preference for rectangularity in the human vision system [11DN (Perkins) ,Visual discrimination between rectangular and nonrectang ular parallelopipeds, Percept Psychophys. 12 (), pp.396–400.[Crossref],,[Google Scholar],12DN (Perkins) ,Compensating for distortion in viewing pictures obliquely,Percept Psychophys. 14 (1973), pp.13–18.[Crossref],[Web of Science ®],[Google Scholar]]. As a result, the upper edge appears to be the plane curvea(x) when seen in the directionv(1) , and appears to beb(x) when seen in the direction(v)(2) . This method can be used to construct an ambiguous cylinder that has the two desired appearancesa(x) and (b) (x) when it is seen from the special viewing directionsV(1) and(v)(2) ,

The surface thus constructed isx– monotone and hence is not closed. If we want a closed cylinder, we can apply the above method twice, once for the upper half of the cylinder and once more for the lower half.

4. How to make topology-disturbing objects

We have reviewed a method for constructing a cylindrical surface whose upper edge has two desired appearances when it is viewed from two special directions. We can use this method to construct topology-disturbing objects, in the following way:

Consider the object shown inFigure 1;Figure 8shows the shape of the sections of the object that we see. From the direct view, we perceive two non-intersecting rectangles as shown in (a), and from the mirror image, we perceive two intersecting rectangles as shown in (b). We decompose the shape in (b) into two non-intersecting closed curves as shown in (c), where, for simplicity, the two closed curves are displaced so that they do not touch. We apply the method for ambiguous cylinders to the curves in (a) and (c). That is, we construct two ambiguous cylinders, one for the upper pair and one for the lower pair in (Figure 8(a, c). Thus, we obtain two cylinders, each of which has the desired appearance.

Figure 8.Desired appearance of two rectangular cylinders: (a) direct view; (b) view in the mirror; (c) decomposition of (b) into two non-intersecting shapes.

The remaining problem is to combine the two cylinders so that they appear to be separated when viewed from the first directionV1) , and they appear to touch when viewed from the second view directionv(2) ; note that if we move the curves in Figure 8(c) so that they touch, they will appear as in (b), which will be perceived as intersecting. For this purpose, we place the two cylinders in different vertical positions as shown inFigure 9. The two cylinders are placed apart, and then their vertical positions are adjusted so that they appear to touch when seen from one of the viewing directions, as shown by the broken line in the figure. We will discuss later how to place the cylinders in more detail. From this method, we obtain a topology-disturbing object.

Figure 16.General view of the object shown in Figure 4.

For the object shown inFigure 5, we obtain the shape diagram shown inFigure 17; (a) shows the direct view, (b) shows the decomposed pair of non-intersecting curves and (c) shows the mirror image. If we apply our method to (b) and (c), we obtain the object shown in (Figure 5) . A general view of the object is shown inFigure 18.

Figure 17.Perceived shapes of the object shown in Figure 5.

Figure 18.General view of the object shown in Figure 5.

Three other examples of topology-disturbing objects are shown inFigures–21. In each figure, (a) shows the direct view and its image in a mirror, and (b) shows a general view of the object.

Figure 19.Five cylinders with disturbed topology.

Figure 20.Two parallel planes and a cylinder with disturbed topology.

Figure 21.Four cylinders with disturbed topology.

Figure 19shows an object composed of five cylinders. In the direct view, the three cylinders in the front row are touching, and the two cylinders in the back row are touching, but the two rows are separated. However, in the mirror image, all five cylinders intersect the adjacent cylinders.

Figure 20shows an object composed of a cylinder and two parallel planes. In the direct view, both the planes cut through the cylinder, but in the mirror view, they are on opposite sides and separated from the cylinder.

Figure 21shows another object. In the direct view, it consists of two intersecting lens-shaped cylinders, but in the image, they change to two pairs of touching but non-intersecting circular cylinders.

5. Constructability condition

We have seen the construction process of topology-disturbing objects through examples. Next, we consider a general condition under which a real 3D object can be constructed from a given pair of appearances.

LetFbe a line drawing composed of curves drawn on theXY−plane, where each curve is a non-self-intersecting closed curve such as a circle / rectangle or an open curve such as a line segment. We assume that each curve inFis not self-intersecting while different curves may intersect. We decompose the curves inFinto (x) – monotone segments and represent them by equations (Y)=f(1) (x),y=f₂(x),…, (y)=(f)_n( (x) ) in such a way that

(1) two segments do not cross each other (although they may touch), and

(2) an upper segment has a smaller segment number than the lower segment, ie, (f)_i( (x) ) ≥f_J(x) impliesi

For example, suppose thatFis the line drawing composed of two cir cles and a line segment shown inFigure 22(a). This drawing can be decomposed into fivex– monotone segmentsf(1) ( (x) ),f_{2) (x),…, (f) (5) ( (x) ) as shown in (b), where we slightly displaced the horizontal positions of curves so that the touching segments are separated for the convenience of understanding.}

Figure 22.Line drawing and its decomposition intox-monotone segments: (a) first line drawingF; (b) decomposition ofF; (c) second line drawing, which is consistent with (b); (d) another decomposition ofF; (e) third line drawing, which is consistent with (d); (f) fourth line drawing, which is not consistent withF.

Similarly, letGbe another line drawing obtained fromFby replacing each curve in the direction parallel to they−axis, and (y)=g(1) (x), (y)=g_{2) (x),…, (Y)=g(M)(x) arex– monotone segments obtained by decomposingGsatisfying (1) and (2). We are interested in whether we can construct a cylindrical object whose appearances from two special viewpoints coincide with the drawings (F) and (G) . We can prove the next theorem.}

Theorem 5.1:

We can construct a topology-disturbing object whose appearances coincide with the drawingsFandG, if the following conditions are satisfied:

(3)M=n, and

(4) for eachi,f_i(x) andg_(i)(x) span the same (x) range; ie, their leftmost points have the samex−coordinate, and their rightmost points also have the samex−coordinate.

A rough sketch of the proof is as follows. Suppose that the conditions (3) and (4) are satisfied. Then, we can construct the ambiguous cylinders corresponding tof_(i)(x) and (g)_i( (x) ) fori=1, 2,…, (n) . Let the resulting ambiguous cylinders beH_i,i=1, 2,…,n. Next, place the vertical cylindersH(1) , (H) (2) ,…, (H)_nSo that their ap pearance coincides with the drawingFwhen they are seen along the first view direction. This is always possible, because the curvesf(1) (x), (f) (2) (x),…,f_n(x) do not cross each other (see condition (1 )). If we see this collection of the cylinders along the second view direction, each cylinder has the desired appearance specified by the second line drawingG, but their mutual positions may not coincide with (G) . Therefore, as a final step, we translate the cylinders in the direction parallel to the first view direction so that the relative positions in the second appearance coincide withG. Note that the appearance in the first view direction is not changed by these translations because they are in that direction. Note also that these translations can be done without collision because their order in they−direction is the same (see condition (2)). Thus, the theemem can be proved.

Let us go back to the example inFigure 22. Suppose that the second drawingGis given as in (c). This drawing can be decomposed into fivex– monotone segments by cutting the circles at the leftmost and the rightmost points. This decomposition ofGtogether with the decomposition ofFinto (b) satisfies the conditions of the theorem, and hence we can construct the associated topology-disturbing object.

The decomposition of the drawing inFigure 22(a) is not unique. Another decomposition is shown in (d). This decomposition and the unique decomposition of the line drawing in (e) satisfy the conditions in the theorem, and hence we can construct a topology-disturbing object corresponding to the drawings (a) and (b).

Figure 22(f) shows still another drawing composed of the same two circles and the line segment. However, we cannot decompose the drawing (a) so that it is consistent with (f) in terms of condition (4). Hence, we cannot construct a topology-disturbing object associated with the drawings (a) and (f).

6. Concluding remarks

We have presented a class of illusory objects, called topology-disturbing objects, which, when viewed from two special directions, appear to have different topologies; this gives us the impression that they are physically impossible. Therefore, this can be regarded as a new class of ‘impossible objects’. Two viewing directions can be realized simultaneously by using a mirror, and hence, these objects can be displayed effectively (such as for an exhibition) by using a mirror.

These objects may be used in many contexts. In vision science, they offer new material for research on seeing; for example, in order to understand human vision, we must clarify why we easily perceive topological inconsistencies. In science education, these objects could be used to stimulate children to start thinking about visual perception. At least upon one’s first encounter with topology-disturbing objects, they are surprising and mysterious, and thus their unusual visual effects might be used in arts and entertainment. One aspect of our future work is to investigate the possibilities in these directions.

Acknowledgements

The author expresses his sincere thanks to the anonymous reviewers for their valuable comments, by which he could improve the manuscript.

No potential conflict of interest was reported by the author.