Symmetry · Lesson 2 of 7
Making Symmetry by Folding and Cutting
“Predict and explain the mirror patterns made by ink, holes and cuts in folded paper.”
• Explain why pressing folded paper creates matching ink patterns. • Predict reflected positions of punched holes. • Infer one or more fold lines from an unfolded pattern. • Predict the outline produced by cuts through folded layers. • Make square openings in two orientations and verify their properties.
Create a mirror pair instead of searching for one
When you fold a plain sheet, the fold brings matching positions together. This makes folding a way to create a symmetric design, not merely test an existing one. A mark or cut passing through the folded layers appears at related positions when the sheet is opened. Predicting these positions is the central idea in this lesson.
First fold a sheet once and open it. Put a little washable paint on one side, press the halves together and open them again. The paint transfers to the touching places on the other half. In an ideal print, the crease is a mirror line because every transferred patch has its partner across it. Real paint may smudge or fail to transfer completely, so use the intended matching arrangement when discussing the geometry.
Make two or three small paint patches rather than covering the whole sheet. Before pressing, predict where the patches will appear on the other side. After opening, compare their distances from the crease and their orientations. Test the crease as an axis. Another axis is possible only if the particular completed print happens to match across it too.
Problem
A small round paint patch is 3 grid squares to the left of a vertical crease. Where should its reflected copy appear?
- 1.The vertical crease acts as a mirror. Move straight across it rather than up or down.
- 2.The copy lies 3 grid squares to the right, at the same height. The crease is halfway between the corresponding positions.
- 3.For a round patch, the outline looks unchanged. For an arrow or an irregular patch, the transferred outline is reversed as a mirror image; it must not simply be copied with the same left-right orientation.
A punch passes through several layers
Punching a small hole through folded paper produces one hole in each layer that the punch passes through. When the sheet is unfolded, these holes are related by the folds. Start with one fold away from the hole, then investigate two folds that cross. A hole on a crease is a special case because reflected positions can coincide.
Problem
A square sheet is folded along its vertical midline. A hole is punched away from the crease. What appears when the sheet is opened?
- 1.The punch passes through two layers, so opening the sheet separates the two punched positions.
- 2.The holes have equal distances from the vertical crease and the same height. One is the reflection of the other.
- 3.The line through the midpoint between the holes, at right angles to the joining segment, identifies the fold direction. In this case it is the vertical midline.
For a diagonal fold, the matching pair may appear at different heights as well as different horizontal positions. The correct comparison is made straight across the diagonal. Turning your drawing so that the axis appears vertical can make that comparison easier, but turning the page does not change which points are partners.
Problem
An unfolded square has one hole near each corner, with equal horizontal and vertical offsets from its centre. How could one punch make this pattern?
- 1.Fold the square along its vertical midline so that left and right positions meet.
- 2.Fold the resulting rectangle along its horizontal midline so that top and bottom positions also meet. The punch now passes through four layers.
- 3.After unfolding, the single punched position produces four holes: one in each reflected quadrant. Reversing the order of the two perpendicular folds gives the same arrangement.
Do not automatically double the number of distinct holes for every fold. The simple two-hole and four-hole counts assume that the punch is away from all fold lines and reaches the relevant layers. A point on a mirror line is fixed, so some reflected positions may coincide.
Predict a cut before opening the paper
A cut along a folded edge creates an opening whose other part is revealed when the sheet is unfolded. Imagine reflecting each part of the cut boundary across the crease. If there are two folds, continue the reflection across both. This predicts the outline without relying on whether the small folded cut resembles the final opening.
Try a V-shaped notch in the folded edge. Opening the paper supplies a second V facing the first. An uneven zigzag creates a matched partner of that zigzag, not necessarily a familiar polygon. Fold a rectangular strip several times to make linked decorative repeats, and inspect which creases still serve as axes of the whole strip. Repeated pieces do not guarantee that every line between pieces is an axis.
Problem
A rectangular sheet is folded vertically. A sloping straight cut joins two points on the folded corner, at equal distances along perpendicular edges. What shape can the opening form?
- 1.Look at the triangular corner removed from the folded layers. The two equal distances make its sloping edge balanced relative to the two edges.
- 2.Unfolding across the appropriate creases reflects that edge to supply the remaining equal sides.
- 3.With vertical and horizontal folds through the centre, equal intercepts give a turned square opening. If the two distances are unequal, the opening may look like a diamond but does not have the right angles needed for a square.
Draw an irregular notch on a folded edge and sketch the expected unfolded opening before using scissors. Repeat with a zigzag, then with two perpendicular folds. Compare each prediction with the opened sheet. Explain each mismatch by locating the boundary segment whose reflected partner you placed incorrectly.
A square can appear in two orientations
A square does not stop being a square when it is turned. Its defining checks are four equal sides and four right angles. The two openings below use different fold directions to produce these properties. A single straight cut is enough when the paper layers supply the other sides through reflection.
Problem
How can two perpendicular centre folds create the opening that looks like a diamond?
- 1.Fold a square sheet in half vertically and horizontally. The original centre is now a corner of the folded packet where both folded edges meet.
- 2.Measure equal short distances from that corner along the two folded edges. Join those two points and make one straight cut along the joining segment.
- 3.When the sheet is unfolded, the four reflected cut segments form a turned square. The equal distances ensure equal sides and right angles. Check those properties rather than naming any four-sided opening a square.
Problem
How can a square opening parallel to the outside edges be made with one straight cut?
- 1.Fold the square along both of its diagonal mirror lines. The original centre again becomes a corner of the folded packet.
- 2.Choose equal distances along the two folded diagonal edges and cut straight across the corner between those points.
- 3.Unfold both diagonal creases. The four reflected segments form a square whose sides are parallel to the original sheet’s sides. The choice of fold axes controls the orientation of the resulting opening.
The punch and cut activities are reverse problems as well. Given an opened pattern, look for a line that pairs every hole or cut edge with a partner. Several different folding procedures can sometimes create the same final figure. A good explanation states a valid procedure and checks the whole result, instead of assuming that the appearance reveals one unique history.
Quiz
Why does an ideal ink-blot print have a mirror line at its crease?
A point is 4 squares left of a vertical fold and away from the fold. Where is its partner?
What usually results from one punch away from both perpendicular creases through a four-layer packet?
What must be checked before calling a diamond-shaped opening a square?
Which method is useful for predicting an unfolded cut?
Why might punching directly on a crease give fewer distinct hole positions?
The fold brings reflected positions together, so pressing creates matching partners.
Practice Problems
- Design an ink-blot outline with exactly one intended mirror line. Explain how you will avoid accidentally creating a second one.
- Draw a vertical fold and a proposed hole 2 squares to its right. Add the matching hole and explain the distances.
- Draw a diagonal fold and two reflected holes. Explain how turning the page helps you check the pair.
- Give a valid two-fold procedure that creates four holes near the corners from a single punch. Explain the layer count.
- Make a zigzag cut in a folded edge. Predict the entire opened boundary before unfolding.
- Explain why unequal distances along two perpendicular folded edges need not produce a square opening.
- Make the two square openings in the diagram, using one straight cut for each. Check both the side lengths and the angles.
- Create a decorative folded-paper strip. Identify the mirror lines of the complete strip and distinguish them from lines that only match two neighbouring parts.
Use the vertical and horizontal midlines. Punch away from both creases through all four layers. The opened holes occupy the four reflected positions.
Key Takeaways
• Folding brings reflected positions together and can generate symmetric patterns. • A punch or cut appears in all the layers it reaches. • Predict opened shapes by reflecting their complete cut boundaries. • Two perpendicular folds can create four matching hole positions. • An opening is a square only when its equal-side and right-angle properties both hold.