This research project sought to find general track formations that allowed battery operated locomotives to traverse the entire train track in both directions infinitely. These formations allowed for any number of track pieces from TrackMaster Thomas and Friends sets by Fisher Price. The research looked at start position of the train and presetting of the switches as well as what is necessary to have a complete track with no dead ends. Surprisingly, there was found to be only one track formation that allowed for entire traversal of the track in both directions. This layout was termed a  dog bone  and consisted of two switch pieces connected at the ends with the tips of both switches connecting to themselves on the same switch. A proof that this layout is the only layout that satisfies the conditions is given

Hamilton, Seth

Struss, Breanna M

English

Valparaiso University

Valparaiso UniversityValpoScholarFall Interdisciplinary Research Symposium Office of Sponsored and Undergraduate ResearchSummer 2015Designing a Self-reversing Track Layout withTrackMaster TracksBreanna M. StrussValparaiso University, breanna.struss@valpo.eduSeth HamiltonValparaiso University, seth.hamilton@valpo.eduFollow this and additional works at: http://scholar.valpo.edu/firesThis Poster Presentation is brought to you for free and open access by the Office of Sponsored and Undergraduate Research at ValpoScholar. It has beenaccepted for inclusion in Fall Interdisciplinary Research Symposium by an authorized administrator of ValpoScholar. For more information, pleasecontact a ValpoScholar staff member at scholar@valpo.edu.Recommended CitationStruss, Breanna M. and Hamilton, Seth, "Designing a Self-reversing Track Layout with TrackMaster Tracks" (2015). FallInterdisciplinary Research Symposium. Paper 38.http://scholar.valpo.edu/fires/38Designing a Self-reversing Track Layout With TrackMasterTM TracksSeth Hamilton and Breanna StrussAdvisor: Dr. James CaristiAbstractThis research project sought to find general track formations that al-lowed battery operated locomotives to traverse the entire train trackin both directions infinitely. These formations allowed for any num-ber of track pieces from TrackMasterTM Thomas and FriendsTM setsby Fisher Price R©. The research looked at start position of the trainand presetting of the switches as well as what is necessary to have acomplete track with no dead ends. Surprisingly, there was found tobe only one track formation that allowed for entire traversal of thetrack in both directions. This layout was termed a “dog bone” andconsisted of two switch pieces connected at the ends with the tipsof both switches connecting to themselves on the same switch. Aproof that this layout is the only layout that satisfies the conditionsis given.FoundationsHere are some simple definitions that will build a foundation forthe latter portions of the article.•Definition 1.1 An end is an entry point to the switch piecesuch that when the train enters the end, it can travel in one oftwo directions but cannot change the switch’s state (part c inFigure 5). A tip is an entry point to the switch piece such thata train entering the switch from a tip must exit through the end,and the train could change the switch’s state (parts a and b inFigure 5).•Definition 1.2 A self-reversing track is a track layout in whicha train infinitely traverses part of the track in opposite directions.•Definition 1.3 A dog bone is a track layout in which two switchpieces are connected to each other at the ends, and the tips of thesame switch are connected. (see Figure 3) It can be seen thatthe dog bone may be extended and made more “complicated.”•Definition 1.4 An arc is any connection between a tip andanother tip of the same or different switch piece. (*note a tip toan end is not an arc and an end to an end is not an arc)•The initial conditions of the track layout such as the position ofthe train and switches, as well as the direction of the train haveno ability to make a layout that doesn’t work a layout that works.SwitchTrainStartAB CDFigure 1• In this figure, depending on the initial conditions, a train willtraverse one dog bone out of 4 possibilities. The train can traversea dog bone between arcs A and D, A and C, B and D, or Band C.•Definition 1.5 Dead track is track that is not infinitelytraversed.•Definition 1.6 A closed track is a track layout such thatfor every piece of track, every entry point must be connectedto another entry point. (*note: A track that is not closed willhave dead track)•Definition 1.7 A switch is in position 0 if it is all the wayleft as viewed from the end of the switch. A switch is in posi-tion 1 if it is all the way right as viewed from the end of theswitch.a bcDiagram of a Switch in Position 0a bcDiagram of a Switch in Position 1Figure 2Why the dog bone is the only layout that works1st Traversal2nd TraversalSwitchStartFigure 3Theorem Any track layout that is self-reversing with no dead track is a dog-bone.Lemma: If a train enters a finite set of tracks, S, at an entry point, x and exits S at an entry point, y, and proceedsdirectly to x without entering any part of S, then the train will continually exit S at y.Proof.· of LemmaConsider Figure 4. Assume a train enters a set of track, S, at x and exits S at y one time. Let P be the first switch piece inS where x is connected to P . During the first traversal, if P connects to x at an end, then the train cannot re-enter P withoutviolating the assumption of the Lemma. If P connects to x at a tip, then the train must exit P through the end. Thus, thetrain will traverse the same path through the switch piece P in both the first and the second traversal, because the switch’sstate will not be changed. Now, consider Figure 5. Let x′ be the exit point of the switch P that is used in both the first andthe second traversals. Let S ′ be the set of all track in S connected to x′ and y. This is the same as Figure 4 with one lessswitch piece. We now have a situation in which x′ acts as x and the train still exits S ′ at y. This argument can be appliedrecursively for every switch piece in S until there are no switch pieces left, and thus the train will infinitely traverse from x toy the same way each time.SxyFigure 4x x′yP S ′Figure 5Corollary In any track layout with no dead track, the end of a switch must connect to the end of another switch.If a tip is connected to an end, then when the train exits at the tip, it will pass through the end and re-exit the switch at thetip, in which case the Lemma applies. Similarly, if the train exits at an end and re-enters the tip, it is forced to exit throughthe end, and the Lemma applies again. In both cases there is dead track due to the other tip that is never traversed. Thisviolates the assumption. Therefore, an end can only be connected to another end.Proof of the TheoremProof. Assume we have a self-reversing track layout with no deadtrack.By the Corollary, we know that a track layout with no dead trackmust have the ends of every switch piece connected to another endof a different switch piece. Consider Figure 6 where a, b, c, and dare tips and R and T are switch pieces. If the train traverses froma tip of R to a tip of T , the train will infinitely traverse the trackin a continuous loop because the switches cannot change their stateand the Lemma applies. This results in dead track. ConsiderFigure 7. If the train exits at the tip, a, of R and re-enters at a,then tip b must also connect to itself, because we know that a tipfrom R cannot connect to a tip on T . So, when both tips connectto themselves, there is dead track since either a or b will not bereachable due to the train’s inability to change the state of theswitch. Therefore, the only possible connection is for the tips of Rto arc to each other and for the tips on T to do the same. Thus, wehave two switches that connect to each other at the ends and haveself-arcing tips. Therefore, any track layout that allows continualtraversal of the track in both directions, with no dead track is infact the dog bone layout as described in Definition 1.3, with twoswitches connected at their ends and self-arcing tips on both switchpieces.SwitchabcdRTFigure 6SwitchRTFigure 7If you want to know more about this topic, please email us atbreanna.struss@valpo.edu and seth.hamilton@valpo.edu.

Designing a Self-reversing Track Layout with TrackMaster Tracks

ValpoScholar

Valparaiso University
ValpoScholar
Symposium on Undergraduate Research and
Creative Expression (SOURCE) Office of Sponsored and Undergraduate Research
Summer 7-28-2015
Designing a Self-reversing Track Layout with
TrackMaster Tracks
Breanna M. Struss
Valparaiso University, breanna.struss@valpo.edu
Seth Hamilton
Valparaiso University, seth.hamilton@valpo.edu
Follow this and additional works at: https://scholar.valpo.edu/cus
This Poster Presentation is brought to you for free and open access by the Office of Sponsored and Undergraduate Research at ValpoScholar. It has been
accepted for inclusion in Symposium on Undergraduate Research and Creative Expression (SOURCE) by an authorized administrator of ValpoScholar.
For more information, please contact a ValpoScholar staff member at scholar@valpo.edu.
Recommended Citation
Struss, Breanna M. and Hamilton, Seth, "Designing a Self-reversing Track Layout with TrackMaster Tracks" (2015). Symposium on
Undergraduate Research and Creative Expression (SOURCE). 477.
https://scholar.valpo.edu/cus/477
Designing a Self-reversing Track Layout With TrackMasterTM Tracks
Seth Hamilton and Breanna Struss
Advisor: Dr. James Caristi
Abstract
This research project sought to find general track formations that al-
lowed battery operated locomotives to traverse the entire train track
in both directions infinitely. These formations allowed for any num-
ber of track pieces from TrackMasterTM Thomas and FriendsTM sets
by Fisher Price R©. The research looked at start position of the train
and presetting of the switches as well as what is necessary to have a
complete track with no dead ends. Surprisingly, there was found to
be only one track formation that allowed for entire traversal of the
track in both directions. This layout was termed a “dog bone” and
consisted of two switch pieces connected at the ends with the tips
of both switches connecting to themselves on the same switch. A
proof that this layout is the only layout that satisfies the conditions
is given.
Foundations
Here are some simple definitions that will build a foundation for
the latter portions of the article.
•Definition 1.1 An end is an entry point to the switch piece
such that when the train enters the end, it can travel in one of
two directions but cannot change the switch’s state (part c in
Figure 5). A tip is an entry point to the switch piece such that
a train entering the switch from a tip must exit through the end,
and the train could change the switch’s state (parts a and b in
Figure 5).
•Definition 1.2 A self-reversing track is a track layout in which
a train infinitely traverses part of the track in opposite directions.
•Definition 1.3 A dog bone is a track layout in which two switch
pieces are connected to each other at the ends, and the tips of the
same switch are connected. (see Figure 2) It can be seen that
the dog bone may be extended and made more “complicated.”
•Definition 1.4 An arc is any connection between a tip and
another tip of the same or different switch piece. (*note a tip to
an end is not an arc and an end to an end is not an arc)
•The initial conditions of the track layout such as the position of
the train and switches, as well as the direction of the train have
no ability to make a layout that doesn’t work a layout that works.
Switch
Train
Start
A
B C
D
Figure 6
• In this figure, depending on the initial conditions, a train will
traverse one dog bone out of 4 possibilities. The train can traverse
a dog bone between arcs A and D, A and C, B and D, or B
and C.
•Definition 1.5 Dead track is track that is not infinitely
traversed.
•Definition 1.6 A closed track is a track layout such that
for every piece of track, every entry point must be connected
to another entry point. (*note: A track that is not closed will
have dead track)
•Definition 1.7 A switch is in position 0 if it is all the way
left as viewed from the end of the switch. A switch is in posi-
tion 1 if it is all the way right as viewed from the end of the
switch.
a b
c
Diagram of a Switch in Position 0
a b
c
Diagram of a Switch in Position 1
Figure 5
Why the dog bone is the only layout that works
1st Traversal
2nd Traversal
Switch
Start
Figure 2
Theorem Any track layout that is self-reversing with no dead track is a dog-bone.
Lemma: If a train enters a finite set of tracks, S, at an entry point, x and exits S at an entry point, y, and proceeds
directly to x without entering any part of S, then the train will continually exit S at y.
Proof.· of Lemma
Consider Figure 8. Assume a train enters a set of track, S, at x and exits S at y one time. Let P be the first switch piece in
S where x is connected to P . During the first traversal, if P connects to x at an end, then the train cannot re-enter P without
violating the assumption of the Lemma. If P connects to x at a tip, then the train must exit P through the end. Thus, the
train will traverse the same path through the switch piece P in both the first and the second traversal, because the switch’s
state will not be changed. Now, consider Figure 9. Let x′ be the exit point of the switch P that is used in both the first and
the second traversals. Let S ′ be the set of all track in S connected to x′ and y. This is the same as Figure 8 with one less
switch piece. We now have a situation in which x′ acts as x and the train still exits S ′ at y. This argument can be applied
recursively for every switch piece in S until there are no switch pieces left, and thus the train will infinitely traverse from x to
y the same way each time.
S
x
y
Figure 8
x x′
y
P S ′
Figure 9
Corollary In any track layout with no dead track, the end of a switch must connect to the end of another switch.
If a tip is connected to an end, then when the train exits at the tip, it will pass through the end and re-exit the switch at the
tip, in which case the Lemma applies. Similarly, if the train exits at an end and re-enters the tip, it is forced to exit through
the end, and the Lemma applies again. In both cases there is dead track due to the other tip that is never traversed. This
violates the assumption. Therefore, an end can only be connected to another end.
Proof of the Theorem
Proof. Assume we have a self-reversing track layout with no dead
track.
By the Corollary, we know that a track layout with no dead track
must have the ends of every switch piece connected to another end
of a different switch piece. Consider Figure 10 where a, b, c, and
d are tips and R and T are switch pieces. If the train traverses from
a tip of R to a tip of T , the train will infinitely traverse the track
in a continuous loop because the switches cannot change their state
and the Lemma applies. This results in dead track. Consider
Figure 11. If the train exits at the tip, a, of R and re-enters
at a, then tip b must also connect to itself, because we know that
a tip from R cannot connect to a tip on T . So, when both tips
connect to themselves, there is dead track since either a or b will
not be reachable due to the train’s inability to change the state of
the switch. Therefore, the only possible connection is for the tips
of R to arc to each other and for the tips on T to do the same.
Thus, we have two switches that connect to each other at the ends
and have self-arcing tips. Therefore, any track layout that allows
continual traversal of the track in both directions, with no dead
track is in fact the dog bone layout as described in Definition
1.4, with two switches connected at their ends and self-arcing tips
on both switch pieces.
Switch
a
bc
d
RT
Figure 10
Switch
RT
Figure 11
If you want to know more about this topic, please email us at
breanna.struss@valpo.edu and seth.hamilton@valpo.edu.


Valparaiso University
ValpoScholar
Fall Interdisciplinary Research Symposium Office of Sponsored and Undergraduate Research
Summer 2015
Designing a Self-reversing Track Layout with
TrackMaster Tracks
Breanna M. Struss
Valparaiso University, breanna.struss@valpo.edu
Seth Hamilton
Valparaiso University, seth.hamilton@valpo.edu
Follow this and additional works at: http://scholar.valpo.edu/fires
This Poster Presentation is brought to you for free and open access by the Office of Sponsored and Undergraduate Research at ValpoScholar. It has been
accepted for inclusion in Fall Interdisciplinary Research Symposium by an authorized administrator of ValpoScholar. For more information, please
contact a ValpoScholar staff member at scholar@valpo.edu.
Recommended Citation
Struss, Breanna M. and Hamilton, Seth, "Designing a Self-reversing Track Layout with TrackMaster Tracks" (2015). Fall
Interdisciplinary Research Symposium. Paper 38.
http://scholar.valpo.edu/fires/38
Designing a Self-reversing Track Layout With TrackMasterTM Tracks
Seth Hamilton and Breanna Struss
Advisor: Dr. James Caristi
Abstract
This research project sought to find general track formations that al-
lowed battery operated locomotives to traverse the entire train track
in both directions infinitely. These formations allowed for any num-
ber of track pieces from TrackMasterTM Thomas and FriendsTM sets
by Fisher Price R©. The research looked at start position of the train
and presetting of the switches as well as what is necessary to have a
complete track with no dead ends. Surprisingly, there was found to
be only one track formation that allowed for entire traversal of the
track in both directions. This layout was termed a “dog bone” and
consisted of two switch pieces connected at the ends with the tips
of both switches connecting to themselves on the same switch. A
proof that this layout is the only layout that satisfies the conditions
is given.
Foundations
Here are some simple definitions that will build a foundation for
the latter portions of the article.
•Definition 1.1 An end is an entry point to the switch piece
such that when the train enters the end, it can travel in one of
two directions but cannot change the switch’s state (part c in
Figure 5). A tip is an entry point to the switch piece such that
a train entering the switch from a tip must exit through the end,
and the train could change the switch’s state (parts a and b in
Figure 5).
•Definition 1.2 A self-reversing track is a track layout in which
a train infinitely traverses part of the track in opposite directions.
•Definition 1.3 A dog bone is a track layout in which two switch
pieces are connected to each other at the ends, and the tips of the
same switch are connected. (see Figure 3) It can be seen that
the dog bone may be extended and made more “complicated.”
•Definition 1.4 An arc is any connection between a tip and
another tip of the same or different switch piece. (*note a tip to
an end is not an arc and an end to an end is not an arc)
•The initial conditions of the track layout such as the position of
the train and switches, as well as the direction of the train have
no ability to make a layout that doesn’t work a layout that works.
Switch
Train
Start
A
B C
D
Figure 1
• In this figure, depending on the initial conditions, a train will
traverse one dog bone out of 4 possibilities. The train can traverse
a dog bone between arcs A and D, A and C, B and D, or B
and C.
•Definition 1.5 Dead track is track that is not infinitely
traversed.
•Definition 1.6 A closed track is a track layout such that
for every piece of track, every entry point must be connected
to another entry point. (*note: A track that is not closed will
have dead track)
•Definition 1.7 A switch is in position 0 if it is all the way
left as viewed from the end of the switch. A switch is in posi-
tion 1 if it is all the way right as viewed from the end of the
switch.
a b
c
Diagram of a Switch in Position 0
a b
c
Diagram of a Switch in Position 1
Figure 2
Why the dog bone is the only layout that works
1st Traversal
2nd Traversal
Switch
Start
Figure 3
Theorem Any track layout that is self-reversing with no dead track is a dog-bone.
Lemma: If a train enters a finite set of tracks, S, at an entry point, x and exits S at an entry point, y, and proceeds
directly to x without entering any part of S, then the train will continually exit S at y.
Proof.· of Lemma
Consider Figure 4. Assume a train enters a set of track, S, at x and exits S at y one time. Let P be the first switch piece in
S where x is connected to P . During the first traversal, if P connects to x at an end, then the train cannot re-enter P without
violating the assumption of the Lemma. If P connects to x at a tip, then the train must exit P through the end. Thus, the
train will traverse the same path through the switch piece P in both the first and the second traversal, because the switch’s
state will not be changed. Now, consider Figure 5. Let x′ be the exit point of the switch P that is used in both the first and
the second traversals. Let S ′ be the set of all track in S connected to x′ and y. This is the same as Figure 4 with one less
switch piece. We now have a situation in which x′ acts as x and the train still exits S ′ at y. This argument can be applied
recursively for every switch piece in S until there are no switch pieces left, and thus the train will infinitely traverse from x to
y the same way each time.
S
x
y
Figure 4
x x′
y
P S ′
Figure 5
Corollary In any track layout with no dead track, the end of a switch must connect to the end of another switch.
If a tip is connected to an end, then when the train exits at the tip, it will pass through the end and re-exit the switch at the
tip, in which case the Lemma applies. Similarly, if the train exits at an end and re-enters the tip, it is forced to exit through
the end, and the Lemma applies again. In both cases there is dead track due to the other tip that is never traversed. This
violates the assumption. Therefore, an end can only be connected to another end.
Proof of the Theorem
Proof. Assume we have a self-reversing track layout with no dead
track.
By the Corollary, we know that a track layout with no dead track
must have the ends of every switch piece connected to another end
of a different switch piece. Consider Figure 6 where a, b, c, and d
are tips and R and T are switch pieces. If the train traverses from
a tip of R to a tip of T , the train will infinitely traverse the track
in a continuous loop because the switches cannot change their state
and the Lemma applies. This results in dead track. Consider
Figure 7. If the train exits at the tip, a, of R and re-enters at a,
then tip b must also connect to itself, because we know that a tip
from R cannot connect to a tip on T . So, when both tips connect
to themselves, there is dead track since either a or b will not be
reachable due to the train’s inability to change the state of the
switch. Therefore, the only possible connection is for the tips of R
to arc to each other and for the tips on T to do the same. Thus, we
have two switches that connect to each other at the ends and have
self-arcing tips. Therefore, any track layout that allows continual
traversal of the track in both directions, with no dead track is in
fact the dog bone layout as described in Definition 1.3, with two
switches connected at their ends and self-arcing tips on both switch
pieces.
Switch
a
bc
d
RT
Figure 6
Switch
RT
Figure 7
If you want to know more about this topic, please email us at
breanna.struss@valpo.edu and seth.hamilton@valpo.edu.


Valparaiso UniversityValpoScholarSymposium on Undergraduate Research andCreative Expression (SOURCE) Office of Sponsored and Undergraduate ResearchSummer 7-28-2015Designing a Self-reversing Track Layout withTrackMaster TracksBreanna M. StrussValparaiso University, breanna.struss@valpo.eduSeth HamiltonValparaiso University, seth.hamilton@valpo.eduFollow this and additional works at: https://scholar.valpo.edu/cusThis Poster Presentation is brought to you for free and open access by the Office of Sponsored and Undergraduate Research at ValpoScholar. It has beenaccepted for inclusion in Symposium on Undergraduate Research and Creative Expression (SOURCE) by an authorized administrator of ValpoScholar.For more information, please contact a ValpoScholar staff member at scholar@valpo.edu.Recommended CitationStruss, Breanna M. and Hamilton, Seth, "Designing a Self-reversing Track Layout with TrackMaster Tracks" (2015). Symposium onUndergraduate Research and Creative Expression (SOURCE). 477.https://scholar.valpo.edu/cus/477Designing a Self-reversing Track Layout With TrackMasterTM TracksSeth Hamilton and Breanna StrussAdvisor: Dr. James CaristiAbstractThis research project sought to find general track formations that al-lowed battery operated locomotives to traverse the entire train trackin both directions infinitely. These formations allowed for any num-ber of track pieces from TrackMasterTM Thomas and FriendsTM setsby Fisher Price R©. The research looked at start position of the trainand presetting of the switches as well as what is necessary to have acomplete track with no dead ends. Surprisingly, there was found tobe only one track formation that allowed for entire traversal of thetrack in both directions. This layout was termed a “dog bone” andconsisted of two switch pieces connected at the ends with the tipsof both switches connecting to themselves on the same switch. Aproof that this layout is the only layout that satisfies the conditionsis given.FoundationsHere are some simple definitions that will build a foundation forthe latter portions of the article.•Definition 1.1 An end is an entry point to the switch piecesuch that when the train enters the end, it can travel in one oftwo directions but cannot change the switch’s state (part c inFigure 5). A tip is an entry point to the switch piece such thata train entering the switch from a tip must exit through the end,and the train could change the switch’s state (parts a and b inFigure 5).•Definition 1.2 A self-reversing track is a track layout in whicha train infinitely traverses part of the track in opposite directions.•Definition 1.3 A dog bone is a track layout in which two switchpieces are connected to each other at the ends, and the tips of thesame switch are connected. (see Figure 2) It can be seen thatthe dog bone may be extended and made more “complicated.”•Definition 1.4 An arc is any connection between a tip andanother tip of the same or different switch piece. (*note a tip toan end is not an arc and an end to an end is not an arc)•The initial conditions of the track layout such as the position ofthe train and switches, as well as the direction of the train haveno ability to make a layout that doesn’t work a layout that works.SwitchTrainStartAB CDFigure 6• In this figure, depending on the initial conditions, a train willtraverse one dog bone out of 4 possibilities. The train can traversea dog bone between arcs A and D, A and C, B and D, or Band C.•Definition 1.5 Dead track is track that is not infinitelytraversed.•Definition 1.6 A closed track is a track layout such thatfor every piece of track, every entry point must be connectedto another entry point. (*note: A track that is not closed willhave dead track)•Definition 1.7 A switch is in position 0 if it is all the wayleft as viewed from the end of the switch. A switch is in posi-tion 1 if it is all the way right as viewed from the end of theswitch.a bcDiagram of a Switch in Position 0a bcDiagram of a Switch in Position 1Figure 5Why the dog bone is the only layout that works1st Traversal2nd TraversalSwitchStartFigure 2Theorem Any track layout that is self-reversing with no dead track is a dog-bone.Lemma: If a train enters a finite set of tracks, S, at an entry point, x and exits S at an entry point, y, and proceedsdirectly to x without entering any part of S, then the train will continually exit S at y.Proof.· of LemmaConsider Figure 8. Assume a train enters a set of track, S, at x and exits S at y one time. Let P be the first switch piece inS where x is connected to P . During the first traversal, if P connects to x at an end, then the train cannot re-enter P withoutviolating the assumption of the Lemma. If P connects to x at a tip, then the train must exit P through the end. Thus, thetrain will traverse the same path through the switch piece P in both the first and the second traversal, because the switch’sstate will not be changed. Now, consider Figure 9. Let x′ be the exit point of the switch P that is used in both the first andthe second traversals. Let S ′ be the set of all track in S connected to x′ and y. This is the same as Figure 8 with one lessswitch piece. We now have a situation in which x′ acts as x and the train still exits S ′ at y. This argument can be appliedrecursively for every switch piece in S until there are no switch pieces left, and thus the train will infinitely traverse from x toy the same way each time.SxyFigure 8x x′yP S ′Figure 9Corollary In any track layout with no dead track, the end of a switch must connect to the end of another switch.If a tip is connected to an end, then when the train exits at the tip, it will pass through the end and re-exit the switch at thetip, in which case the Lemma applies. Similarly, if the train exits at an end and re-enters the tip, it is forced to exit throughthe end, and the Lemma applies again. In both cases there is dead track due to the other tip that is never traversed. Thisviolates the assumption. Therefore, an end can only be connected to another end.Proof of the TheoremProof. Assume we have a self-reversing track layout with no deadtrack.By the Corollary, we know that a track layout with no dead trackmust have the ends of every switch piece connected to another endof a different switch piece. Consider Figure 10 where a, b, c, andd are tips and R and T are switch pieces. If the train traverses froma tip of R to a tip of T , the train will infinitely traverse the trackin a continuous loop because the switches cannot change their stateand the Lemma applies. This results in dead track. ConsiderFigure 11. If the train exits at the tip, a, of R and re-entersat a, then tip b must also connect to itself, because we know thata tip from R cannot connect to a tip on T . So, when both tipsconnect to themselves, there is dead track since either a or b willnot be reachable due to the train’s inability to change the state ofthe switch. Therefore, the only possible connection is for the tipsof R to arc to each other and for the tips on T to do the same.Thus, we have two switches that connect to each other at the endsand have self-arcing tips. Therefore, any track layout that allowscontinual traversal of the track in both directions, with no deadtrack is in fact the dog bone layout as described in Definition1.4, with two switches connected at their ends and self-arcing tipson both switch pieces.SwitchabcdRTFigure 10SwitchRTFigure 11If you want to know more about this topic, please email us atbreanna.struss@valpo.edu and seth.hamilton@valpo.edu.

https://scholar.valpo.edu/cgi/viewcontent.cgi?article=1495&amp;context=cus

Designing a Self-reversing Track Layout with TrackMaster Tracks

Abstract

Similar works

Full text

Available Versions

Valparaiso University

ValpoScholar

ValpoScholar

Valparaiso University